nLab geometry of physics -- categories and toposes

Redirected from "geometry of physics -- Categories and Toposes".
Contents

This entry is one chapter of geometry of physics.

next chapters: smooth sets, supergeometry

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Context

Category theory

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Category theory and topos theory concern the general abstract structure underlying algebra, geometry and logic. They are ubiquituous in and indispensible for organizing conceptual mathematical frameworks.

We give here an introduction to the basic concepts and results, aimed at providing background for the synthetic higher supergeometry of relevance in formulations of fundamental physics, such as used in the chapters on perturbative quantum field theory and on fundamental super p-branes. For quick informal survey see Introduction to Higher Supergeometry.

This makes use of the following curious dictionary between category theory/topos theory and the geometry of generalized spaces, which we will explain in detail (following Grothendieck 65, Lawvere 86, p. 17, Lawvere 91):

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A\phantom{A}category theoryRmk. A\phantom{A}geometry of generalized spaces
A\phantom{A}presheafExpl. A\phantom{A}generalized space
A\phantom{A}representable presheafA\phantom{A}Expl. A\phantom{A}A\phantom{A}model space
A\phantom{A}regarded as generalized space
A\phantom{A}Yoneda lemmaProp. A\phantom{A}A\phantom{A}sets of probes of generalized spaces
A\phantom{A}are indeed
A\phantom{A}sets of maps from model spaces A\phantom{A}
A\phantom{A}Yoneda embedding A\phantom{A}Prop. A\phantom{A}A\phantom{A}nature of model spaces is preserved when
A\phantom{A}regarding them as generalized spaces A\phantom{A}
A\phantom{A}Yoneda embedding isA\phantom{A}
A\phantom{A}free co-completionA\phantom{A}
Prop. A\phantom{A}generalized spaces really areA\phantom{A}
A\phantom{A}glued from ordinary spacesA\phantom{A}
A\phantom{A}topos theoryRmk. A\phantom{A}local-global principle for generalized spacesA\phantom{A}
A\phantom{A}coverageDefn. A\phantom{A}notion of locality
A\phantom{A}sheaf conditionDefn. A\phantom{A}
Prop.
A\phantom{A}plots of generalized spaces
A\phantom{A}satisfy local-to-global principle A\phantom{A}
A\phantom{A}comparison lemmaProp. A\phantom{A}notion of generalized spaces
A\phantom{A}independent under change of model space
A\phantom{A}gros topos theoryRmk. A\phantom{A}generalized spaces at the foundations
A\phantom{A}cohesionDefn. A\phantom{A}generalized spaces obey
A\phantom{A}principles of differential topology
A\phantom{A}differential cohesionDefn. A\phantom{A}generalized spaces obey
A\phantom{A}principles of differential geometry
A\phantom{A}super cohesionA\phantom{A}Defn. A\phantom{A}generalized spaces obey
A\phantom{A}principles of supergeometry

The perspective is that of functorial geometry (Grothendieck 65). (For more exposition of this point see also at motivation for sheaves, cohomology and higher stacks.) This dictionary implies a wealth of useful tools for handling and reasoning about geometry:

We discuss below that sheaf toposes, regarded as categories of generalized spaces via the above disctionary, are “convenient contexts” for geometry (Prop. below), in the technical sense that they provide just the right kind of generalization that makes all desireable constructions on spaces actually exist:

A\phantom{A}sheaf toposA\phantom{A}A\phantom{A}as category of generalized spaces A\phantom{A}
A\phantom{A}Yoneda embedding: A\phantom{A}A\phantom{A}contains and generalizes ordinary spaces A\phantom{A}
A\phantom{A}has all limits: A\phantom{A}A\phantom{A}contains all Cartesian products and intersections A\phantom{A}
A\phantom{A}has all colimits: A\phantom{A}A\phantom{A}contains all disjoint unions and quotients
A\phantom{A}cartesian closure: A\phantom{A}A\phantom{A}contains all mapping spacesA\phantom{A}
A\phantom{A}local cartesian closure: A\phantom{A}A\phantom{A}contains all fiber-wise mapping spaces A\phantom{A}

Notably mapping spaces play a pivotal role in physics, in the guise of spaces of field histories, but fall outside the applicability of traditional formulations of geometry based on just manifolds. Topos theory provides their existence (Prop. below) and the relevant infrastructure, for example for the construction of transgression of differential forms to mapping spaces of smooth sets, that is the basis for sigma-model-field theories. This is discussed in the following chapters on smooth sets and on supergeometry.

In conclusion, one motivation for category theory and topos theory is a posteriori: As a matter of experience, there is just no other toolbox that allows to deeply understand and handle the geometry of physics. Similar comments apply to a wealth of other topics of mathematics.

We offer also an a priori motivation:

Category theory is the theory of duality.

Duality is of course an ancient notion in philosophy. At least as a term, it makes a curious re-appearance in the conjectural theory of fundamental physics formerly known as string theory, as duality in string theory. In both cases, the literature left some room in delineating what precisely is meant. But the philosophically inclined mathematician could notice (see Lambek 82) that an excellent candidate to make precise the idea of duality is the mathematical concept of adjunction, from category theory. This is particularly pronounced for adjoint triples (Remark below) and their induced adjoint modalities (Lawvere 91, see Def. below), which exhibit a given “mode of being” of any object XX as intermediate between two dual opposite extremes (Prop. below):

XAAAAXAAAAX \Box X \overset{\phantom{AAAA}}{\longrightarrow} X \overset{\phantom{AAAA}}{\longrightarrow} \bigcirc X

For example, cohesive geometric structure on generalized spaces is captured, this way, as modality in between the discrete and the codiscrete (Example , and Def. below).

Historically, category theory was introduced in order to make precise the concept of natural transformation: The concept of functors was introduced just so as to support that of natural transformations, and the concept of categories only served that of functors (see Freyd 1964, p. 1).

But natural transformations are, in turn, exactly the basis for the concept of adjoint functors (Def. below), equivalently adjunctions between categories (Prop. below), shown on the left. All universal constructions, the heart of category theory, are special cases of adjoint functors – hence of dualities, if we follow Lambek 82: This includes the concepts of limits and colimits (Def. below), ends and coends (Def. below) Kan extensions (Prop. below), and the behaviour of these constructions, such as for instance the free co-completion nature of the Yoneda embedding (Prop. below).

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Therefore it makes sense to regard category theory as the theory of adjunctions,
hence the theory of duality:


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A\phantom{A}hierarchy of conceptsA\phantom{A}A\phantom{A}category theoryA\phantom{A}A\phantom{A}enrichedA\phantom{A}A\phantom{A}homotopicalA\phantom{A}
A\phantom{A}adjunction of adjunctionsA\phantom{A}
AA\phantom{AA}duality of dualitiesA\phantom{A}
A\phantom{A}Def. A\phantom{A}A\phantom{A}Def. A\phantom{A}
A\phantom{A} adjoint equivalenceA\phantom{A}
AA\phantom{AA}dual equivalence AA\phantom{AA}
A\phantom{A} Def. A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A}Def.
A\phantom{A} adjunctionA\phantom{A}
AA\phantom{AA}dualityA\phantom{A}
A\phantom{A} Def. A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A}Def.
A\phantom{A} natural transformation A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A} Def. A\phantom{A}
A\phantom{A} functor A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A} Def. A\phantom{A}
A\phantom{A} category A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A} Def. A\phantom{A}A\phantom{A} Def.

The pivotal role of adjunctions in category theory (Lawvere 08) and in the foundations of mathematics (Lawvere 69, Lawvere 94 ) was particularly amplified by F. W. Lawvere1. Moreover, Lawvere saw the future of category theory (Lawvere 91) as concerned with adjunctions expressing systems of archetypical dualities that reveal foundations for geometry (Lawvere 07) and physics (Lawvere 97, see Def. and Def. below). He suggested (Lawvere 94) this as a precise formulation of core aspects of the theory of everything of early 19th century philosophy: Hegel‘s Science of Logic.

These days, of course, theories of everything, such as string theory, are understood less ambitiously than Hegel’s ontological process, as mathematical formulations of fundamental theories of physics, that could conceptually unify the hodge-podge of currently available “standard models” of particle physics and of cosmology to a more coherent whole.

The idea of duality in string theory refers to different perspectives on physics that appear dual to each other while being equivalent. But one of the basic results of category theory (Prop. , below) is that equivalence is indeed a special case of adjunction. This allows to explore the possibility that there is more than a coincidence of terms.

Of course the usage of the term duality in string theory is too loose for one to expect to be able to refine each occurrence of the term in the literature to a mathematical adjunction. However, we will see mathematical formalizations of core aspects of key string-theoretic dualities, such as topological T-duality and the duality between M-theory and type IIA string theory, in terms of adjunctions. Indeed, at the heart of these dualities in string theory is the phenomenon of double dimensional reduction, which turns out to be formalized by one of the most fundamental adjunctions in (higher) category theory: base change along the point inclusion into a classifying space. All this is discussed in the chapter on fundamental super p-branes.

This suggests that there may be a deeper relation here between the superficially alien uses of the word “duality”, that is worth exploring.

In this respect it is worth noticing that core structure of string/M-theory arises via universal constructions from the superpoint (as explained in the chapter on fundamental super p-branes), while the superpoint itself arises, in a sense made precise by category theory, “from nothing”, by a system of twelve adjunctions (explained in the chapter on supergeometry).

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Here we introduce the requisites for understanding these statements.

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Contents

Basic notions of Category theory

We introduce here the basic notions of category theory, along with examples and motivation from geometry:

  1. Categories and functors

  2. Natural transformations and presheaves

  3. Adjunctions

  4. Equivalences

  5. Modalities

This constitutes what is sometimes called the language of categories. While we state and prove some basic facts here, notably the notorious Yoneda lemma (Prop. below), what makes category theory be a mathematical theory in the sense of a coherent collection of non-trivial theorems is all concerned with the topic of universal constructions, which may be formulated (only) in this language. This we turn to further below.

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Categories and Functors

The notion of a category (Def. below) embodies the idea of structuralism applied to concepts in mathematics: it collects, on top of the set (or generally: class) of mathematical objects that belong to it, also all the structure-preserving maps between them, hence the homomorphisms in the case of Bourbaki-style mathematical structures.

The first achievement of the notion of a category is to abstract away from such manifestly concrete categories (Examples , below) to more indirectly defined mathematical objects whose “structure” is only defined, after the fact, by which maps, now just called morphisms, there are between them.

This structuralism-principle bootstraps itself to life by considering morphisms between categories themselves to be those “maps” that respect their structuralism, namely the connectivity and composition of the morphisms between their objects: These are the functors (Def. below).

For the purpose of geometry, a key class of examples of functors are the assignments of algebras of functions to spaces, this is Example below.

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Definition

(category)

A category 𝒞\mathcal{C} is

  1. a class Obj 𝒞Obj_{\mathcal{C}}, called the class of objects;

  2. for each pair X,YObj 𝒞X,Y \in Obj_{\mathcal{C}} of objects, a set Hom 𝒞(X,Y)Hom_{\mathcal{C}}(X,Y), called the set of morphisms from XX to YY, or the hom-set, for short.

    We denote the elements of this set by arrows like this:

    XfYHom 𝒞(X,Y). X \overset{f}{\longrightarrow} Y \;\;\in Hom_{\mathcal{C}}(X,Y) \,.
  3. for each object XObj 𝒞X \in Obj_{\mathcal{C}} a morphism

    Xid XXHom 𝒞(X,X) X \overset{id_X}{\to} X \;\; \in Hom_{\mathcal{C}}(X,X)

    called the identity morphism on XX;

  4. for each triple X 1,X 2,X 3ObjX_1, X_2, X_3 \in Obj of objects, a function

    Hom 𝒞(X 1,X 2) × Hom 𝒞(X 2,X 3) X 1,X 2,X 3 Hom 𝒞(X 1,X 3) X 1fX 2 , X 2fX 3 X 1gfX 3 \array{ Hom_{\mathcal{C}}(X_1, X_2) &\times& Hom_{\mathcal{C}}(X_2, X_3) &\overset{\circ_{X_1,X_2,X_3}}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \\ X_1 \overset{f}{\to} X_2 &,& X_2 \overset{f}{\to} X_3 &\mapsto& X_1 \overset{ g \circ f }{\longrightarrow} X_3 }

    called composition;

such that:

  1. for all pairs of objects X,YObj 𝒞X,Y \in Obj_{\mathcal{C}} unitality holds: given

    XfYHom 𝒞(X,Y) X \overset{f}{\to} Y \;\;\in Hom_{\mathcal{C}}(X,Y)

    then

    Xid YfY=XfY=Xfid XY; X \overset{id_Y \circ f}{\longrightarrow} Y \;=\; X \overset{f}{\longrightarrow} Y \;=\; X \overset{f \circ id_X }{\longrightarrow} Y \,;
  2. for all quadruples of objects X 1,X 2,X 3,X 4Obj 𝒞X_1, X_2, X_3, X_4 \in Obj_{\mathcal{C}} composition satifies associativity: given

    X 1f 12X 2f 23X 3f 34X 4 X_1 \overset{f_{12}}{\to} X_2 \overset{f_{23}}{\to} X_3 \overset{f_{34}}{\to} X_4

    then

    X 1f 34(f 23f 12)X 4=X 1(f 34f 23)f 12X 4. X_1 \overset{f_{34} \circ (f_{23} \circ f_{12})}{\longrightarrow} X_4 \;\;=\;\; X_1 \overset{(f_{34} \circ f_{23}) \circ f_{12}}{\longrightarrow} X_4 \,.

The archetypical example of a category is the category of sets:

Example

(category of all sets)

The class of all sets with functions between them is a category (Def. ), to be denoted Set:

  • Obj Set=class of all setsObj_{Set} = \text{class of all sets};

  • Hom Set(X,Y)=set of functions from set X to set YHom_{Set}(X,Y) = \text{set of functions from set X to set Y};

  • id XHom Set(X,X)=id_X \in Hom_{Set}(X,X) = identity function on set XX;

  • X 1,X 2,X 3=ordinary composition of functions\circ_{X_1,X_2,X_3} = \text{ordinary composition of functions}.

More generally all kind of sets with structure, in the sense going back to Bourbaki, form categories, where the morphisms are the homomorphisms (whence the name “morphism”!). These are called concrete categories (we characterize them precisely in Example , further below):

Example

(basic examples of concrete categories)

For 𝒮\mathcal{S} a kind of mathematical structure, there is the category (Def. ) 𝒮Set\mathcal{S}Set whose objects are the corresponding structured sets, and whose morphisms are the corresponding structure homomorphisms, hence the functions of underlying sets which respect the given structure.

Basic examples of concrete categories include the following:

A\phantom{A}concrete categoryA\phantom{A}A\phantom{A}objectsA\phantom{A}A\phantom{A}morphismsA\phantom{A}
A\phantom{A}SetA\phantom{A}setsA\phantom{A}functions
A\phantom{A}TopA\phantom{A}topological spacesA\phantom{A}A\phantom{A}continuous functionsA\phantom{A}
A\phantom{A}Mfd k{}_{k}A\phantom{A}differentiable manifoldsA\phantom{A}A\phantom{A}differentiable functionsA\phantom{A}
A\phantom{A}VectA\phantom{A}vector spacesA\phantom{A}A\phantom{A}linear functionsA\phantom{A}
A\phantom{A}GrpA\phantom{A}groupsA\phantom{A}A\phantom{A}group homomorphismsA\phantom{A}
A\phantom{A}AlgA\phantom{A}algebrasA\phantom{A}A\phantom{A}algebra homomorphismA\phantom{A}

This is the motivation for the terminology “categories”, as the examples in Example are literally categories of mathematical structures. But not all categories are “concrete” in this way.

Some terminology:

Definition

(commuting diagram)

Let 𝒞\mathcal{C} be a category (Def. ), then a directed graph with edges labeled by morphisms of the category is called a commuting diagram if for any two vertices any two ways of passing along edges from one to the other yields the same composition of the corresponding morphisms.

For example, a commuting triangle is

f=hgAAAAAA X g f Y AhA Z f = h \circ g \phantom{AAAAAA} \array{ && X \\ & {}^{\mathllap{ g }}\swarrow && \searrow^{ \mathrlap{ f } } \\ Y && \underset{\phantom{A}h\phantom{A}}{\longrightarrow} && Z }

while a commuting square is

g 1f 1=g 2f 2AAAAAAX Af 1A Y 1 f 2 g 1 Y 2 Ag 2A Z g_1 \circ f_1 \;=\; g_2 \circ f_2 \phantom{AAAAAA} \array{ X &\overset{\phantom{A}f_1\phantom{A}}{\longrightarrow}& Y_1 \\ {}^{ \mathllap{f_2} }\big\downarrow && \big\downarrow^{\mathrlap{ g_1 }} \\ Y_2 &\underset{\phantom{A}g_2\phantom{A}}{\longrightarrow}& Z }
Definition

(initial object and terminal object)

Let 𝒞\mathcal{C} be a category (Def. ). Then

  1. an object *𝒞\ast \in \mathcal{C} is called a terminal object if for every other object c𝒞c \in \mathcal{C}, there is a unique morphism from cc to *\ast

    c!* c \overset{\exists!}{\longrightarrow} \ast

    hence if the hom-set is a singleton *Set\ast \in Set:

    Hom 𝒞(c,*)*. Hom_{\mathcal{C}}(c,\ast) \;\simeq\; \ast \,.
  2. an object 𝒞\emptyset \in \mathcal{C} is called an initial object if for every other object c𝒞c \in \mathcal{C}, there is a unique morphism from \emptyset to cc

    !c \emptyset \overset{\exists!}{\longrightarrow} c

    hence if the hom-set is a singleton *Set\ast \in Set:

    Hom 𝒞(,c)*. Hom_{\mathcal{C}}(\emptyset,c) \;\simeq\; \ast \,.
Definition

(small category)

If a category 𝒞\mathcal{C} (Def. ) happens to have as class Obj 𝒞Obj_{\mathcal{C}} of objects an actual set (i.e. a small set instead of a proper class), then 𝒞\mathcal{C} is called a small category.

As usual, there are some trivial examples, that are however usefully made explicit for the development of the theory:

Example

(initial category and terminal category)

  1. The terminal category *\ast is the category (Def. ) whose class of objects is the singleton set, and which has a single morphism on this object, necessarily the identity morphism.

  2. The initial category or empty category \emptyset is the category (Def. ) whose class of objects is the empty set, and which, hence, has no morphism whatsoever.

Clearly, these are small categories (Def. ).

Example

(preordered sets as thin categories)

Let (S,)(S, \leq) be a preordered set. Then this induces a small category whose set of objects is SS, and which has precisely one morphism xyx \to y whenever xyx \leq y, and no such morphism otherwise:

(1)x!yAAAprecisely ifAAAxy x \overset{\exists !}{\to} y \phantom{AAA} \text{precisely if} \phantom{AAA} x \leq y

Conversely, every small category with at most one morphism from any object to any other, called a thin category, induces on its set of objects the structure of a partially ordered set via (1).

Here the axioms for preordered sets and for categories match as follows:

A\phantom{A}reflexivityA\phantom{A}A\phantom{A}transitivityA\phantom{A}
A\phantom{A}partially ordered setsA\phantom{A}A\phantom{A} xxx \leq x A\phantom{A}A\phantom{A} (xyz)(xz)(x \leq y \leq z) \Rightarrow (x \leq z) A\phantom{A}
A\phantom{A}thin categoriesA\phantom{A}A\phantom{A}identity morphismsA\phantom{A}A\phantom{A}compositionA\phantom{A}
Definition

(isomorphism)

For 𝒞\mathcal{C} a category (Def. ), a morphism

XfYHom 𝒞(X,Y) X \overset{f}{\to} Y \;\;\in Hom_{\mathcal{C}}(X,Y)

is called an isomorphism if there exists an inverse morphism

Yf 1XHom 𝒞(Y,X) Y \overset{f^{-1}}{\longrightarrow} X \;\; \in Hom_{\mathcal{C}}(Y,X)

namely a morphism such that the compositions with ff are equal to the identity morphisms on XX and YY, respectively

f 1f=id XAAAff 1=id Y f^{-1} \circ f \;=\; id_X \phantom{AAA} f \circ f^{-1} \;=\; id_Y
Definition

(groupoid)

If 𝒞\mathcal{C} is a category in which every morphism is an isomorphism (Def. ), then 𝒞\mathcal{C} is called a groupoid.

Example

(delooping groupoid)

For GG a group, there is a groupoid (Def. ) BG\mathbf{B}G with a single object, whose single hom-set is GG, with identity morphism the neutral element and composition the group operation in GG:

  • Obj BG=*Obj_{\mathbf{B}G} = \ast

  • Hom 𝒞(*,*)=GHom_{\mathcal{C}}(\ast,\ast) \;=\; G

In fact every groupoid with precisely one object is of the form.

Remark

(groupoids and homotopy theory)

Even though groupoids (Def. ) are special cases of categories (Def. ), the theory of groupoids in itself has a rather different flavour than that of category theory: Part of the homotopy hypothesis-theorem is that the theory of groupoids is really homotopy theory for the special case of homotopy 1-types.

(In applications in homotopy theory, groupoids are considered mostly in the case that the class Obj 𝒞Obj_{\mathcal{C}} of objects is in fact a set: small groupoids, Def. ).

For this reason we will not have more to say about groupoids here, and instead relegate their discussion to the section on homotopy theory, further below.

There is a range of constructions that provide new categories from given ones:

Example

(opposite category and formal duality)

Let 𝒞\mathcal{C} be a category. Then its opposite category 𝒞 op\mathcal{C}^{op} has the same objects as 𝒞\mathcal{C}, but the direction of the morphisms is reversed. Accordingly, composition in the opposite category 𝒞 op\mathcal{C}^{op} is that in 𝒞\mathcal{C}, but with the order of the arguments reversed:

  • Obj 𝒞 opObj 𝒞Obj_{\mathcal{C}^{op}} \;\coloneqq\; Obj_{\mathcal{C}};

  • Hom 𝒞 op(X,Y)Hom 𝒞(Y,X)Hom_{\mathcal{C}^{op}}(X,Y) \;\coloneqq\; Hom_{\mathcal{C}}(Y,X).

Hence for every statementa about some category 𝒞\mathcal{C} there is a corresponding “dual” statement about its opposite category, which is “the same but with the direction of all morphisms reversed”. This relation is known as formal duality.

Example

(product category)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two categories (Def. ). Then their product category 𝒞×𝒟\mathcal{C} \times \mathcal{D} has as objects pairs (c,d)(c,d) with cObj 𝒞c \in Obj_{\mathcal{C}} and dObj 𝒟d \in Obj_{\mathcal{D}}, and as morphisms pairs (c 1fc 2)Hom 𝒞(c 1,c 2) (c_1 \overset{f}{\to} c_2) \in Hom_{\mathcal{C}}(c_1,c_2), (d 1gd 2)Hom 𝒟(d 1,d 2) (d_1 \overset{g}{\to} d_2) \in Hom_{\mathcal{D}}(d_1,d_2), and composition is defined by composition in each entry:

  • Obj 𝒞×𝒟Obj 𝒞×Obj 𝒟Obj_{\mathcal{C} \times \mathcal{D}} \coloneqq Obj_{\mathcal{C}} \times Obj_{\mathcal{D}};

  • Hom 𝒞×𝒟((c 1,d 1),(c 2,d 2))Hom 𝒞(c 1,c 2)×Hom 𝒟(d 1,d 2)Hom_{\mathcal{C} \times \mathcal{D}}( (c_1,d_1), (c_2,d_2) ) \coloneqq Hom_{\mathcal{C}}(c_1,c_2) \times Hom_{\mathcal{D}}( d_1, d_2 )

  • (f 2,g 2)(f 1,g 1)(f 2f 1,g 2g 1)(f_2, g_2) \circ (f_1, g_1) \;\coloneqq\; ( f_2 \circ f_1, g_2 \circ g_1 )

Definition

(functor)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two categories (Def. ). A functor from 𝒞\mathcal{C} to 𝒟\mathcal{D}, to be denoted

𝒞F𝒟 \mathcal{C} \overset{F}{\longrightarrow} \mathcal{D}

is

  1. a function between the classes of objects:

    F Obj:Obj 𝒞Obj 𝒟 F_{Obj} \;\colon\; Obj_{\mathcal{C}} \longrightarrow Obj_{\mathcal{D}}
  2. for each pair X,YObj 𝒞X,Y \in Obj_{\mathcal{C}} of objects a function

    F X,Y:Hom 𝒞(X,Y)Hom 𝒟(F Obj(X),F Obj(Y)) F_{X,Y} \;\colon\; Hom_{\mathcal{C}}(X,Y) \longrightarrow Hom_{\mathcal{D}}(F_{Obj}(X), F_{Obj}(Y))

such that

  1. For each object XObj 𝒞X \in Obj_{\mathcal{C}} the identity morphism is respected:

    F X,X(id X)=id F Obj(X); F_{X,X}(id_X) \;=\; id_{F_{Obj}(X)} \,;
  2. for each triple X 1,X 2,X 3Obj 𝒞X_1, X_2, X_3 \in Obj_{\mathcal{C}} of objects, composition is respected: given

    X 1fX 2gX 3 X_1 \overset{f}{\longrightarrow} X_2 \overset{g}{\longrightarrow} X_3

    we have

    F X 1,X 3(gf)=F X 2,X 3(g)F X 1,X 2(f). F_{X_1, X_3}(g \circ f ) \;=\; F_{X_2, X_3}(g) \circ F_{X_1,X_2}(f) \,.
Example

(categories of small categories and of small groupoids)

It is clear that functors (Def. ) have a composition operation given componentwise by the composition of their component functions. Accordingly, this composition is unital and associative. This means that there is

  1. the category (Def. ) Cat whose objects are small categories (Def. ) and whose morphisms are functors (Def. ) between these

  2. the category (Def. ) Grpd whose objects are small groupoids (Def. ) and whose morphisms are functors (Def. ) between these.

Example

(hom-functor)

Let 𝒞\mathcal{C} be a category (Def. ). Then its hom-functor

Hom 𝒞:𝒞 op×𝒞Set Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set

is the functor (Def. ) out of the product category (Def. ) of 𝒞\mathcal{C} with its opposite category to the category of sets, which sends a pair X,Y𝒞X,Y \in \mathcal{C} of objects to the hom-set Hom 𝒞(X,Y)Hom_{\mathcal{C}}(X,Y) between them, and which sends a pair of morphisms, with one of them into XX and the other out of YY, to the operation of composition with these morphisms:

Hom 𝒞:X 1 Y 1 g h X 2 Y 2Hom 𝒞(X 1,Y 1) fhfg Hom 𝒞(X 2,Y 2) Hom_{\mathcal{C}} \;\;\colon\;\;\; \array{ X_1 & Y_1 \\ {}^{\mathllap{g}}\big\uparrow & \big\downarrow^{\mathrlap{h}} \\ X_2 & Y_2 } \;\;\mapsto\;\; \array{ Hom_{\mathcal{C}}(X_1, Y_1) \\ \big\downarrow^{ \mathrlap{ f \mapsto h \circ f \circ g } } \\ Hom_{\mathcal{C}}(X_2, Y_2) }
Definition

(monomorphism and epimorphism)

Let 𝒞\mathcal{C} be a category (Def. ). Then a morphism XfYX \overset{f}{\to } Y in 𝒞\mathcal{C} is called

  • a monomorphism if for every object Z𝒞Z \in \mathcal{C} the hom-functor (Example ) out of ZZ takes ff to an injective function of hom-sets:

    Hom 𝒞(Z,f):Hom 𝒞(Z,X)AAAHom 𝒞(Z,Y); Hom_{\mathcal{C}}(Z,f) \;\colon\; Hom_{\mathcal{C}}(Z,X) \overset{\phantom{AAA}}{\hookrightarrow} Hom_{\mathcal{C}}(Z,Y) \,;
  • an epimorphism if for every object Z𝒵Z \in \mathcal{Z} the hom-functor (Example ) into ZZ takes ff to an injective function:

    Hom 𝒞(f,Z):Hom 𝒞(Y,Z)AAAHom 𝒞(X,Z). Hom_{\mathcal{C}}( f,Z ) \;\colon\; Hom_{\mathcal{C}}(Y, Z) \overset{\phantom{AAA}}{\hookrightarrow} Hom_{\mathcal{C}}(X, Z) \,.
Definition

(full, faithful and fully faithful functors)

A functor F:𝒞𝒟F \;\colon\; \mathcal{C} \to \mathcal{D} (Def. ) is called

  • a full functor if all its hom-functions are surjective functions

    Hom 𝒞(X,Y)surjF X,YHom 𝒟(F(X),F(Y)) Hom_{\mathcal{C}}(X,Y) \underoverset{surj}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))
  • a faithful functor if all its hom-functions are injective functions

    Hom 𝒞(X,Y)injF X,YHom 𝒟(F(X),F(Y)) Hom_{\mathcal{C}}(X,Y) \underoverset{inj}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))
  • a fully faithful functor if all its hom-functions are bijective functions

    Hom 𝒞(X,Y)bijF X,YHom 𝒟(F(X),F(Y)) Hom_{\mathcal{C}}(X,Y) \underoverset{bij}{F_{X,Y}}{\longrightarrow} Hom_{\mathcal{D}}(F(X), F(Y))

A fully faithful functor is also called a full subcategory-inclusion. We will denote this situation by

𝒞AFA𝒟. \mathcal{C} \overset{\phantom{A}F\phantom{A}}{\hookrightarrow} \mathcal{D} \,.
Example

(full subcategory on a sub-class of objects)

Let 𝒞\mathcal{C} be a category (Def. ) and let SObj 𝒞S \subset Obj_{\mathcal{C}} be a sub-class of its class of objects. The there is a category 𝒞 S\mathcal{C}_S whose class of objects is SS, and whose morphisms are precisely the morphisms of 𝒞\mathcal{C}, between these given objects:

Hom 𝒞 S(s 1,s 2)Hom 𝒞(s 1,s 2) Hom_{\mathcal{C}_S}(s_1, s_2) \;\coloneqq\; Hom_{\mathcal{C}}(s_1, s_2)

with identity morphisms and composition defined as in 𝒞\mathcal{C}. Then there is a fully faithful functor (Def. )

𝒞 S AAAA 𝒞 \array{ \mathcal{C}_S &\overset{\phantom{AAAA}}{\hookrightarrow}& \mathcal{C} }

which is the evident inclsuion on objects, and the identity function on all hom-sets.

This is called the full subcategory of 𝒞\mathcal{C} on the objects in SS.

Beware that not every fully faithful functor is, in components, exactly of this form, but, assuming the axiom of choice, every fully faithful functor is so up to equivalence of categories (Def. ).

The concept of faithful functor from Def. allows to make precise the idea of concrete category from Example :

Example

(structured sets and faithful functors)

Let 𝒮\mathcal{S} be a kind of mathematical structure and let 𝒮Set\mathcal{S} Set be the category of 𝒮\mathcal{S}-structured sets. Then there is the forgetful functor

𝒮SetSet \mathcal{S}Set \longrightarrow Set

which sends each structured set to the underlying set (“forgetting” the structure that it carries), and which sends functions of sets to themselves. That a homomorphism of structured sets is a function between the underlying sets satisfying a special condition implies that this is a faithful functor (Def. ).

Conversely, it makes sense to define structured sets in general to be the objects of a category 𝒞\mathcal{C} which is equipped with a faithful functor 𝒞faithfulSet\mathcal{C} \overset{faithful}{\longrightarrow} Set to the category of sets. See at structure for more on this.

Example

(spaces seen via their algebras of functions)

In any given context of geometry, there is typically a functor which sends any space of the given kind to its algebra of functions, and which sends a map (i.e. homomorphism) between the given spaces to the algebra homomorphism given by precomposition with that map (a hom-functor, Def. ). Schematically:

{geometric spaces} algebra of functions {algebras} op X 1 FunctionsOn(X 1) f ϕϕf X 2 FunctionsOn(X 2) \array{ \big\{ \text{geometric spaces} \big\} & \overset{ \text{algebra of functions} }{ \longrightarrow } & \big\{ \text{algebras} \big\}^{op} \\ \\ X_1 &\mapsto& FunctionsOn(X_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \phi \mapsto \phi \circ f } \\ X_2 &\mapsto& FunctionsOn(X_2) }

Since the precomposition operation reverses the direction of morphisms, as shown, these are functors from the given category of spaces to the opposite (Example ) of the relevant category of algebras.

In broad generality, there is a duality (“Isbell duality”) between geometry/spaces and algebra/algebras of functions) (“space and quantity”, Lawvere 86).

We now mention some concrete examples of this general pattern:

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topological spaces and C*-algebras

Consider

  1. the category Top cpt{}_{cpt} of compact topological Hausdorff spaces with continuous functions between them;

  2. the category C*Alg of unital C*-algebras over the complex numbers

from Example .

Then there is a functor (Def. )

C():Top H,cptC *Alg op C(-) \;\colon\; Top_{H,cpt} \longrightarrow C^\ast Alg^{op}

from the former to the opposite category of the latter (Example ) which sends any compact topological space XX to its C*-algebra C(X)C(X) of continuous functions XϕX \overset{\phi}{\to} \mathbb{C} with values in the complex numbers, and which sends every continuous function between compact spaces to the C*-algebra-homomorphism that is given by precomposition:

C():X C(X) f f *:ϕϕf Y C(Y) C(-) \;\;\;\colon\;\;\; \array{ X &\mapsto & C(X) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow^{\mathrlap{ f^\ast : \phi \mapsto \phi \circ f }} \\ Y &\mapsto& C(Y) }

Part of the statement of Gelfand duality is that this is a fully faithful functor, hence exhibiting a full subcategory-inclusion (Def. ), namely that of commutative C*-algebras:

Top H,cptAAAC *Alg op. Top_{H,cpt} \overset{\phantom{AAA}}{\hookrightarrow} C^\ast Alg^{op} \,.

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affine schemes and commutative algebras

The starting point of algebraic geometry is to consider affine schemes as the formal duals (Example ) of finitely generated commutative algebras over some algebraically closed ground field 𝕂\mathbb{K}:

(2)Aff 𝕂CAlg 𝕂 finop. Aff_{\mathbb{K}} \;\;\coloneqq\;\; CAlg^{fin}_{\mathbb{K}}^{op} \,.

Beware that the immediate identification (2) is often obscured by the definition of affine schemes as locally ringed spaces. While the latter is much more complicated, at face value, in the end it yields an equivalent category (Def. below) to the simple formal dualization (Example ) in (2), see here. Already in 1973 Alexander Grothendieck had urged to abandon, as a foundational concept, the more complicated definition in favor of the simpler one in (2), see Lawvere 03.

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smooth manifolds and real associative algebras

Consider

  1. the category SmthMfd of smooth manifolds with smooth functions between them;

  2. the category Alg {}_{\mathbb{R}} of associative algebras over the real numbers

from Example .

Then there is a functor (Def. )

C ():SmthMfdAlg op C^\infty(-) \;\colon\; SmthMfd \longrightarrow Alg_{\mathbb{R}}^{op}

from the former to the opposite category of the latter (Def. ), which sends any smooth manifold XX to its associative algebra C (X)C^\infty(X) of continuous functions XϕX \overset{\phi}{\to} \mathbb{R} to the real numbers, and which sends every smooth function between smooth manifolds to the algebra homomorphism that is given by precomposition:

C ():X C (X) f f *:ϕϕf Y C (Y) C^\infty(-) \;\;\;\colon\;\;\; \array{ X &\mapsto & C^\infty(X) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow^{\mathrlap{ f^\ast : \phi \mapsto \phi \circ f }} \\ Y &\mapsto& C^\infty(Y) }

The statement of Milnor's exercise is that this this is a fully faithful functor, hence exhibiting a full subcategory-inclusion (Def. ):

SmthMfdAAAAAlg op. SmthMfd \overset{\phantom{AAAA}}{ \hookrightarrow } Alg_{\mathbb{R}}^{op} \,.

These two statements, expressing categories of spaces as full subcategories of opposite categories of categories of algebras, are the starting point for many developments in geometry, such as algebraic geometry, supergeometry, noncommutative geometry and noncommutative topology.

\,

Since a fully faithful functor/full subcategory-embedding 𝒞𝒟\mathcal{C} \hookrightarrow \mathcal{D} exhibits the objects of 𝒟\mathcal{D} as a consistent generalization of the objects of 𝒞\mathcal{C}, one may turn these examples around and define more general kinds of spaces as formal duals (Example ) to certain algebras:

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infinitesimally thickened points and formal Cartesian spaces

The category of infinitesimally thickened points is, by definition, the full subcategory (Example ) of the opposite category (Example ) of that of commutative algebras (Example ) over the real numbers

InfThckPoint AAAA Alg op 𝔻 C (𝔻) V \array{ InfThckPoint &\overset{\phantom{AAAA}}{\hookrightarrow}& Alg_{\mathbb{R}}^{op} \\ \mathbb{D} &\mapsto& C^\infty(\mathbb{D}) \\ && \coloneqq \mathbb{R} \oplus V }

on those with a unique maximal ideal VV which is a finite-dimensional as an \mathbb{R}-vector space and a nilradical: for each aVa \in V there exists nn \in \mathbb{N} such that a n=0a^n = 0.

The category of formal Cartesian spaces is, by definition, the full subcategory (Example ) of the opposite category (Example ) of that of commutative algebras (Example ) over the real numbers

FormalCartSp AAAA Alg op n×𝔻 C ( n×𝔻) C ( n) (V) \array{ FormalCartSp &\overset{\phantom{AAAA}}{\hookrightarrow}& Alg_{\mathbb{R}}^{op} \\ \mathbb{R}^n \times \mathbb{D} &\mapsto& C^\infty(\mathbb{R}^n \times \mathbb{D}) \\ && \coloneqq C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}}(\mathbb{R} \oplus V) }

on those which are tensor products of algebras, of an algebra of smooth functions on a Cartesian space n\mathbb{R}^n, for some nn \in \mathbb{Z}, and the algebra of functions on an infinitesimally thickened point.

Notice that the formal Cartesian spaces n|q\mathbb{R}^{n\vert q} are fully defined by this assignment.

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super points and super Cartesian spaces

The category of super points is by definition, the full subcategory (Example ) of the opposite category (Example ) of that of supercommutative algebras (Example ) over the real numbers

SuperPoint AAAA sCAlg op 0|q Λ q \array{ SuperPoint &\overset{\phantom{AAAA}}{\hookrightarrow}& sCAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^{0\vert q} &\mapsto& \Lambda_q }

on the Grassmann algebras:

Λ q[θ 1,,θ q]/(θ iθ j=θ jθ i)AAAAAq. \Lambda_q \;\coloneqq\; \mathbb{R}[ \theta_1, \cdots, \theta_q ]/( \theta_i \theta_j = - \theta_j \theta_i ) \phantom{AAAAA} q \in \mathbb{N} \,.

More generally, the category of super Cartesian spaces is by definition, the full subcategory

SuperCartSp AAAA sCAlg op n|q C ( n) Λ q \array{ SuperCartSp &\overset{\phantom{AAAA}}{\hookrightarrow}& sCAlg_{\mathbb{R}}^{op} \\ \mathbb{R}^{n\vert q} &\mapsto& C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} \Lambda_q }

on the tensor product of algebras, over \mathbb{R} of the algebra of smooth functions on a Cartesian space, and a Grassmann algebra, as above.

Notice that the super Cartesian spaces n|q\mathbb{R}^{n\vert q} are fully defined by this assignment. We discuss this in more detail in the chapter on supergeometry.

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Natural transformations and presheaves

Given a system of (homo-)morphisms (“transformations”) in some category (Def. )

F XAη XAG X F_X \overset{\phantom{A}\eta_X\phantom{A}}{\longrightarrow} G_X

between objects that depend on some variable XX, hence that are values of functors of XX (Def. ), one says that this is natural, hence a natural transformation (Def. below) if it is compatible with (homo-)morphisms of the variable itself.

These natural transformations are the evident homomorphisms between functors

FAηAG, F \overset{\phantom{A}\eta\phantom{A}}{\longrightarrow} G \,,

and hence there is a category of functors between any two categories (Example below).

A key class of such functor categories are those between an opposite category 𝒞 op\mathcal{C}^{op} and the base category of sets, these are also called categories of presheaves (Example below). It makes good sense (Remark below) to think of these as categories of “generalized objects of 𝒞\mathcal{C}”, a perspective which is made precise by the statement of the Yoneda lemma (Prop. below) and the resulting Yoneda embedding (Prop. below). This innocent-looking lemma is the heart that makes category theory tick.

\,

Definition

(natural transformation and natural isomorphism)

Given two categories 𝒞\mathcal{C} and 𝒟\mathcal{D} (Def. ) and given two functors FF and GG from 𝒞\mathcal{C} to 𝒟\mathcal{D} (Def. ), then a natural transformation from FF to GG

𝒞AAηAAGF𝒟 \mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\eta} \phantom{AA}} \mathcal{D}

is

  • for each object XObj 𝒞X \in Obj_{\mathcal{C}} a morphism

    (3)F(X)η XG(X) F(X) \overset{ \eta_X }{\longrightarrow} G(X)

such that

  • for each morphism XfYX \overset{f}{\longrightarrow} Y we have a commuting square (Def. ) of the form

    (4)η YF(X)=G(Y)η XAAAAAAF(X) η X G(X) F(f) G(f) F(Y) η Y G(Y) \eta_Y\circ F(X) \;=\; G(Y)\circ \eta_X \phantom{AAAAAA} \array{ F(X) &\overset{\eta_X}{\longrightarrow}& G(X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(Y) &\underset{\eta_Y}{\longrightarrow}& G(Y) }

    (sometimes called the naturality square of the natural transformation).

If all the component morphisms η X\eta_X are isomorphisms (Def. ), then the natural transformation η\eta is called a natural isomorphism.

For

𝒞AAηAAGF𝒟AAandAA𝒞AAρAAHG𝒟 \mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\eta} \phantom{AA}} \mathcal{D} \phantom{AA} \text{and} \phantom{AA} \mathcal{C} \underoverset {\underset{H}{\longrightarrow}} {\overset{G}{\longrightarrow}} {\phantom{AA}\Downarrow \mathrlap{\rho} \phantom{AA}} \mathcal{D}

two natural transformations as shown, their composition is the natural transformation

𝒞AρηAAAAHF𝒟 \mathcal{C} \underoverset {\underset{H}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{A}\Downarrow \mathrlap{\rho \circ \eta} \phantom{AAAA}} \mathcal{D}

whose components (3) are the compositions of the components of η\eta and ρ\rho:

(5)(ρη) Xρ Xη XAAAAAF(X) η X G(X) ρ X H(X) F(f) G(f) H(f) F(Y) η Y G(Y) ρ Y H(Y) (\rho \circ \eta)_X \;\coloneqq\; \rho_{X} \circ \eta_X \phantom{AAAAA} \array{ F(X) &\overset{\eta_X}{\longrightarrow}& G(X) &\overset{\rho_X}{\longrightarrow}& H(X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} && \downarrow^{\mathrlap{H(f)}} \\ F(Y) &\underset{\eta_Y}{\longrightarrow}& G(Y) &\underset{\rho_Y}{\longrightarrow}& H(Y) }
Example

(reduction of formal Cartesian spaces)

On the category FormalCartSp of formal Cartesian spaces Example , consider the endofunctor

FormalCartSp AAAA FormalCartSp n×𝔻 n \array{ FormalCartSp &\overset{ \phantom{AA}\Re \phantom{AA} }{\longrightarrow}& FormalCartSp \\ \mathbb{R}^n \times \mathbb{D} &\mapsto& \mathbb{R}^n }

which sends each formal Cartesian space to the underlying ordinary Cartesian space, forgetting the infinitesimally thickened point-factor. Moreover, on morphisms this functor is defined via the retraction

id: n i n×𝔻 r n C ( n) quotient projectioni * C ( n) (RV) ff1r * C ( n) \array{ id \colon & \mathbb{R}^n &\overset{i}{\longrightarrow}& \mathbb{R}^n \times \mathbb{D} &\overset{r}{\longrightarrow}& \mathbb{R}^n \\ & C^\infty(\mathbb{R}^n) &\underoverset{\text{quotient projection}}{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n) \otimes_{\mathbb{R}} (R \oplus V) &\underoverset{f \mapsto f \otimes 1}{r^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n) }

as

C ( n×𝔻) AAA C ( n) i * C ( n×𝔻) f * (f *)i *f *r * f * C ( n×𝔻) C ( n) r * C ( n×𝔻) \array{ C^\infty(\mathbb{R}^n \times \mathbb{D}) &\phantom{AAA}&& C^\infty(\mathbb{R}^n) &\overset{i^\ast}{\longleftarrow}& C^\infty( \mathbb{R}^n \times \mathbb{D} ) \\ {}^{\mathllap{ f^\ast }}\big\uparrow && & {}^{ \mathllap{\Re( f^\ast ) \coloneqq i^\ast \circ f^\ast \circ r^\ast } }\big\uparrow && \big\uparrow^{ \mathrlap{ f^\ast } } \\ C^\infty(\mathbb{R}^{n'} \times {\mathbb{D}}') &&& C^\infty(\mathbb{R}^{n'}) &\overset{r^\ast}{\longrightarrow}& C^\infty( \mathbb{R}^{n'} \times {\mathbb{D}}') }

This is indeed functorial due to the fact that any algebra homomorphism f *f^\ast needs to send nilpotent elements to nilpotent elements, so that the following identity holds:

(6)i *f *=i *f *r *i *. i^\ast \circ f^\ast \;=\; i^\ast \circ f^\ast \circ r^\ast \circ i^\ast \,.

Then there is a natural transformation (Def. ) from this functor to the identity functor

Aη AId \Re \overset{ \phantom{A} \eta^{\Re} \phantom{A} }{\longrightarrow} Id

whose components inject the underlying Cartesian space along the unit point inclusion of the infinitesimally thickened point:

( n×𝔻) n Aη n×𝔻 n×𝔻 C ( n) i * C ( n×𝔻) i *f *r * f * C ( n) i * C ( n×𝔻) \array{ \Re\left( \mathbb{R}^n \times \mathbb{D} \right) \coloneqq & \mathbb{R}^n &\overset{ \phantom{A} \eta^\Re_{\mathbb{R}^n \times \mathbb{D}} }{\longrightarrow}& \mathbb{R}^n \times \mathbb{D} \\ & C^\infty(\mathbb{R}^n) &\overset{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^n \times \mathbb{D}) \\ & {}^{ \mathllap{ i^\ast \circ f^\ast \circ r^\ast } }\big\uparrow && \big\uparrow^{\mathrlap{ f^\ast }} \\ & C^\infty(\mathbb{R}^{n'}) &\overset{i^\ast}{\longleftarrow}& C^\infty(\mathbb{R}^{n'} \times \mathbb{D}') }

The commutativity of this naturality square is again the identity (6).

Example

(functor category)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be categories (Def. ). Then the category of functors between them, to be denoted [𝒞,𝒟][\mathcal{C}, \mathcal{D}], is the category whose objects are the functors 𝒞F𝒟\mathcal{C} \overset{F}{\to} \mathcal{D} (Def. ) and whose morphisms are the natural transformations FηGF \overset{\eta}{\Rightarrow} G between functors (Def. ) and whose composition operation is the composition of natural transformations (5).

Example

(category of presheaves)

Given a category 𝒞\mathcal{C} (Def. ), a functor (Def. ) of the form

F:𝒞 opSet, F \;\colon\; \mathcal{C}^{op} \longrightarrow Set \,,

hence out of the opposite category of 𝒞\mathcal{C} (Def. ), into the category of sets (Example ) is also called a presheaf (for reasons discussed below) on 𝒞\mathcal{C} or over 𝒞\mathcal{C}.

The corresponding functor category (Example )

PSh(𝒞)[𝒞 op,Set] PSh(\mathcal{C}) \;\coloneqq\; [\mathcal{C}^{op}, Set]

is hence called the category of presheaves over 𝒞\mathcal{C}.

Example

(representable presheaves)

Given a category 𝒞\mathcal{C} (Def. ), the hom-functor (Example ) induces the following functor (Def. ) from 𝒞\mathcal{C} to its category of presheaves (Def. ):

(7)y : 𝒞 [𝒞 op,Set] c 1 g c 2 X Hom 𝒞(,X) AA:AA Hom 𝒞(c 1,X) Hom 𝒞(g,X) Hom 𝒞(c 2,X) f Hom 𝒞(,f) Hom 𝒞(c 1,f) Hom 𝒞(c 2,f) Y Hom 𝒞(,Y) AA:AA Hom 𝒞(c 1,Y) Hom 𝒞(g,Y) Hom 𝒞(c 2,Y) \array{ y & \colon & \mathcal{C} &\longrightarrow& [\mathcal{C}^{op}, Set] \\ \\ && && && c_1 &\overset{g}{\longrightarrow}& c_2 \\ && X &\mapsto& Hom_{\mathcal{C}}(-,X) &\phantom{AA}\colon\phantom{AA}& Hom_{\mathcal{C}}(c_1,X) &\overset{Hom_{\mathcal{C}}( g, X ) }{\longleftarrow}& Hom_{\mathcal{C}}(c_2, X) \\ && {}^{\mathllap{ f }}\big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}(-,f) } } && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}( c_1, f ) } } && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}(c_2,f) } } \\ && Y &\mapsto& Hom_{\mathcal{C}}(-,Y) &\phantom{AA}\colon\phantom{AA}& Hom_{\mathcal{C}}(c_1,Y) &\overset{Hom_{\mathcal{C}}( g, Y ) }{\longleftarrow}& Hom_{\mathcal{C}}(c_2, Y) }

The presheaves y(X)Hom 𝒞(,X)y(X) \coloneqq Hom_{\mathcal{C}}(-,X) in the image of this functor are called the representable presheaves and XObj 𝒞X \in Obj_{\mathcal{C}} is called their representing object.

The functor (7) is also called the Yoneda embedding, due to Prop. below.

Remark

(presheaves as generalized spaces)

If a given category 𝒞\mathcal{C} (Def. ) is thought of as a category of spaces of sorts, as those in Example , then it will be most useful to think of the corresponding category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] (Def. ) as a category of generalized spaces probe-able by the test spaces in 𝒞\mathcal{C} (Lawvere 86, p. 17).

Namely, imagine a generalized space X\mathbf{X} which is at least probe-able by spaces in 𝒞\mathcal{C}. This should mean that for each object c𝒞c \in \mathcal{C} there is some set of geometric maps “cXc \to \mathbf{X}”. Here the quotation marks are to warn us that, at this point, X\mathbf{X} is not defined yet; and even if it were, it is not expected to be an object of 𝒞\mathcal{C}, so that, at this point, an actual morphism from cc to X\mathbf{X} is not definable. But we may anyway consider some abstract set

(8)X(c)"=Hom(c,X)" \mathbf{X}(c) \; \text{"=} Hom(c,\mathbf{X})"

whose elements we do want to think of maps (homomorphisms of spaces) from cc to X\mathbf{X}.

That this is indeed consistent, in that we may actually remove the quotation remarks on the right of (8), is the statement of the Yoneda lemma, which we discuss as Prop. below.

A minimum consistency condition for this to make sense (we will consider further conditions later on when we discuss sheaves) is that we may consistently pre-compose the would-be maps from cc to X\mathbf{X} with actual morphisms dfcd \overset{f}{\to} c in 𝒞\mathcal{C}. This means that for every such morphism there should be a function between these sets of would-be maps

c X(c) f X(f)"=()f" d X(d) \array{ c && \mathbf{X}(c) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow{}^{\mathrlap{ \mathbf{X}(f) \, \text{"=}(-)\circ f\text{"}}} \\ d && \mathbf{X}(d) }

which respects composition and identity morphisms. But in summary, this says that what we have defined thereby is actually a presheaf on 𝒞\mathcal{C} (Def. ), namely a functor

X:𝒞 opSet. \mathbf{X} \;\colon\; \mathcal{C}^{op} \longrightarrow Set \,.

For consistency of regarding this presheaf as a presheaf of sets of plots of a generalized space, it ought to be true that every “ordinary space”, hence every object X𝒞X \in \mathcal{C}, is also an example of a “generalized space probe-able by” object of 𝒞\mathcal{C}, since, after all, these are the spaces which may manifestly be probed by objects c𝒞c \in \mathcal{C}, in that morphisms cXc \to X are already defined.

Hence the incarnation of X𝒞X \in \mathcal{C} as a generalized space probe-able by objects of 𝒞\mathcal{C} should be the presheaf Hom 𝒞(,X)Hom_{\mathcal{C}}(-,X), hence the presheaf represented by XX (Example ), via the Yoneda functor (7).

At this point, however, a serious consistency condition arises: The “ordinary spaces” now exist as objects of two different categories: on the one hand there is the original X𝒞X \in \mathcal{C}, on the other hand there is its Yoneda image y(X)[𝒞 op,Set]y(X) \in [\mathcal{C}^{op}, Set] in the category of generalized spaces. Hence we need to know that these two perspectives are compatible, notably that maps XYX \to Y between ordinary spaces are the same whether viewed in 𝒞\mathcal{C} or in the more general context of [𝒞 op,Set][\mathcal{C}^{op}, Set].

That this, too, holds true, is the statement of the Yoneda embedding, which we discuss as Prop. below.

Eventually one will want to impose one more consistency condition, namely that plots are determined by their local behaviour. This is the sheaf condition (Def. below) and is what leads over from category theory to topos theory below.

Proposition

(Yoneda lemma)

Let 𝒞\mathcal{C} be a category (Def. ), X𝒞X \in \mathcal{C} any object, and Y[𝒞 op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] a presheaf over 𝒞\mathcal{C} (Def. ).

Then there is a bijection

Hom [𝒞 op,Set](y(X),(Y)) Y(X) η η X(id X) \array{ Hom_{[\mathcal{C}^{op},Set]}( y(X), \mathbf(Y) ) &\overset{\simeq}{\longrightarrow}& \mathbf{Y}(X) \\ \eta &\mapsto& \eta_X(id_X) }

between the hom-set of the category of presheaves from the presheaf represented by XX (7) to Y\mathbf{Y}, and the set which is assigned by Y\mathbf{Y} to XX.

Proof

By Example , an element in the set on the left is a natural transformation (Def. ) of the form

𝒞 opAAηAAYy(X)Set \mathcal{C}^{op} \underoverset {\underset{\mathbf{Y}}{\longrightarrow}} {\overset{y(X)}{\longrightarrow}} {\phantom{AA} \Downarrow \mathrlap{\eta} \phantom{AA}} Set

hence given by component functions (3)

Hom 𝒞(c,X)η cY(X) Hom_{\mathcal{C}}(c,X) \overset{\eta_c}{\longrightarrow} \mathbf{Y}(X)

for each c𝒞c \in \mathcal{C}. In particular there is the component at c=Xc = X

Hom 𝒞(X,X) η X Y(X) id X η X(id X) \array{ Hom_{\mathcal{C}}(X,X) &\overset{\eta_X}{\longrightarrow}& \mathbf{Y}(X) \\ id_X &\mapsto& \eta_X(id_X) }

and the identity morphism id Xid_X on XX is a canonical element in the set on the left. The statement to be proven is hence equivalently that for every element in Y(X)\mathbf{Y}(X) there is precisely one η\eta such that this element equals η X(id X)\eta_X(id_X).

Now the condition to be satisfied by η\eta is that it makes its naturality squares (4) commute (Def. ). This includes those of the form

id X Hom 𝒞(X,X) η X Y(X) Hom 𝒞(f,X) Y(f) Hom 𝒞(Y,X) η Y Y(Y)AAAA{id X} {η X(id X)} {f} {η Y(f)=Y(f)(η X(id X))} \array{ id_X \in & Hom_{\mathcal{C}}(X,X) &\overset{\eta_X}{\longrightarrow}& \mathbf{Y}(X) \\ & {}^{\mathllap{ Hom_{\mathcal{C}}(f,X) }} \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{Y}(f)}} \\ & Hom_{\mathcal{C}}(Y,X) &\underset{\eta_Y}{\longrightarrow}& \mathbf{Y}(Y) } \phantom{AAAA} \array{ \{id_X\} &\longrightarrow& \{\eta_X(id_X)\} \\ \big\downarrow && \big\downarrow \\ \{f\} &\longrightarrow& \big\{ \eta_Y(f) = \mathbf{Y}(f)( \eta_X(id_X) ) \big\} }

for any morphism

(YfX)Hom 𝒞(Y,X). (Y \overset{f}{\longrightarrow} X) \;\in\; Hom_{\mathcal{C}}(Y,X) \,.

As the diagram chase of elements on the right shows, this commutativity (Def. ) fixes η Y(f)\eta_Y(f) for all Y𝒞Y \in \mathcal{C} and all fHom 𝒞(Y,X)f \in Hom_{\mathcal{C}}(Y,X) uniquely in terms of the element η X(id X)\eta_{X}(id_X).

It remains only to see that there is no condition on the element η X(id X)\eta_X(id_X), hence that with η Y(f)\eta_Y(f) defined this way, the commutativity of all the remaining naturality squares is implies: The general naturality square for a morphism Y 2gY 1Y_2 \overset{g}{\longrightarrow} Y_1 is of the form

Hom 𝒞(Y 1,X) η Y 1 Y(Y 1) Hom 𝒞(g,X) Y(g) Hom 𝒞(Y 2,X) η Y 2 Y(Y 2)AAAA{f 1} {Y(f 1)(η X(id X))} {f 2=f 1g} {Y(f 2)(η X(id X))=Y(g)Y(f 1)(η X(id X))} \array{ & Hom_{\mathcal{C}}(Y_1,X) &\overset{\eta_{Y_1}}{\longrightarrow}& \mathbf{Y}(Y_1) \\ & {}^{\mathllap{ Hom_{\mathcal{C}}(g,X) }} \big\downarrow && \big\downarrow{}^{\mathrlap{\mathbf{Y}(g)}} \\ & Hom_{\mathcal{C}}(Y_2,X) &\underset{\eta_{Y_2}}{\longrightarrow}& \mathbf{Y}(Y_2) } \phantom{AAAA} \array{ \{f_1\} &\longrightarrow& \{ \mathbf{Y}(f_1)( \eta_X(id_X) ) \} \\ \big\downarrow && \big\downarrow \\ \{f_2 = f_1\circ g\} &\longrightarrow& \{\mathbf{Y}(f_2)( \eta_X(id_X) ) = \mathbf{Y}(g) \circ \mathbf{Y}(f_1) ( \eta_X(id_X) ) \} }

As shown on the right, the commutativity of this diagram now follows from the functoriality Y(f 2)=Y(f 1g)\mathbf{Y}(f_2) = \mathbf{Y}(f_1 \circ g) of the presheaf Y\mathbf{Y}.

As a direct corollary, we obtain the statement of the Yoneda embedding:

Proposition

(Yoneda embedding)

The assignment (7) of represented presheaves (Example ) is a fully faithful functor (Def. ), hence exhibits a full subcategory inclusion

y:𝒞 AAAA [𝒞 op,Set] X Hom 𝒞(,X) y \;\;\colon\;\; \array{ \mathcal{C} &\overset{\phantom{AAAA}}{\hookrightarrow}& [\mathcal{C}^{op}, Set] \\ X &\mapsto& Hom_{\mathcal{C}}(-,X) }

of the given category 𝒞\mathcal{C} into its category of presheaves.

Proof

We need to show that for all X 1,X 2Obj 𝒞X_1, X_2 \in Obj_{\mathcal{C}} the function

(9)Hom 𝒞(X 1,X 2) Hom [𝒞 op,Set](Hom 𝒞(,X 1),Hom 𝒞(,X 2)) f Hom 𝒞(,f) \array{ Hom_{\mathcal{C}}(X_1, X_2) &\overset{ }{\longrightarrow}& Hom_{[\mathcal{C}^{op}, Set]} \big( Hom_{\mathcal{C}}(-,X_1) \;,\; Hom_{\mathcal{C}}(-,X_2) \big) \\ f &\mapsto& Hom_{\mathcal{C}}(-,f) }

is a bijection. But the Yoneda lemma (Prop. ) states a bijection the other way around

Hom [𝒞 op,Set](Hom 𝒞(,X 1),Hom 𝒞(,X 2)) Hom 𝒞(,X 2)(X 1) = Hom 𝒞(X 1,X 2) η η X 1(id X 1) Hom 𝒞(,f) Hom 𝒞(X 1,f)(id X 1)=f \array{ Hom_{[\mathcal{C}^{op}, Set]} \big( Hom_{\mathcal{C}}(-,X_1) \;,\; Hom_{\mathcal{C}}(-,X_2) \big) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(-,X_2)(X_1) &=& Hom_{\mathcal{C}}(X_1, X_2) \\ \eta && \mapsto && \eta_{X_1}( id_{X_1} ) \\ Hom_{\mathcal{C}}(-,f) && \mapsto && Hom_{\mathcal{C}}(X_1,f)(id_{X_1}) = f }

and hence it is sufficient to see that this is a left inverse to (9). This follows by inspection, as shown in the third line above.

As a direct corollary we obtain the following alternative characterization of isomorphisms, to be compared with the definition of epimorphisms/monomorphisms in Def. :

Example

(isomorphism via bijection of hom-sets)

Let 𝒞\mathcal{C} be a category (Def. ), let X,YObj XX, Y \in Obj_{X} be a pair of objects, and let XfYHom 𝒞(X,Y)X \overset{f}{\to} Y \;\; \in Hom_{\mathcal{C}}(X,Y) be a morphism between them. Then the following are equivalent:

  1. XfYX \overset{f}{\to} Y is an isomorphism (Def. ),

  2. the hom-functors into and out of ff take values in bijections of hom-sets: i.e. for all objects AObj 𝒞A \in Obj_{\mathcal{C}}, we have

    Hom 𝒞(A,f):Hom 𝒞(A,X)Hom 𝒞(A,Y) Hom_{\mathcal{C}}(A,f) \;\colon\; Hom_{\mathcal{C}}(A,X) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(A,Y)

    and

    Hom 𝒞(f,A):Hom 𝒞(Y,A)Hom 𝒞(X,A) Hom_{\mathcal{C}}(f,A) \;\colon\; Hom_{\mathcal{C}}(Y,A) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(X,A)

\,

Adjunctions

The concepts of categories, functors and natural transformations constitute the “language of categories”. This language now allows to formulate the concept of adjoint functors (Def. ) and more generally that of adjunctions (Def. below. This is concept that category theory, as a theory, is all about.

Part of the data involved in an adjunction is its adjunction unit and adjunction counit (Def. below) and depending on their behaviour special cases of adjunctions are identified (Prop. below), which we discuss in detail in following sections:

A\phantom{A}adjunctionA\phantom{A}
A\phantom{A}Def. , Def. A\phantom{A}
A\phantom{A}unit is iso:A\phantom{A}
A\phantom{A}coreflectionA\phantom{A}
A\phantom{A}Def. A\phantom{A}
A\phantom{A}counit is iso:A\phantom{A}A\phantom{A}reflectionA\phantom{A}
A\phantom{A}Def.
A\phantom{A}adjoint equivalenceA\phantom{A}
A\phantom{A}Def. A\phantom{A}

We now discuss four equivalent definitions of adjoint functors:

  1. via hom-isomorphism (Def. below);

  2. via adjunction unit and -counit satisfying triangle identities (Prop. );

  3. via representing objects (Prop. );

  4. via universal morphisms (Prop. below).

Then we discuss some key properties:

  1. uniqueness of adjoints (Prop. below),

  2. epi/mono/iso-characterization of adjunction (co-)units (Prop. below).

\,

Definition

(adjoint functors)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two categories (Def. ), and let

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}

be a pair of functors between them (Def. ), as shown. Then this is called a pair of adjoint functors (or an adjoint pair of functors) with LL left adjoint and RR right adjoint, denoted

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}

if there exists a natural isomorphism (Def. ) between the hom-functors (Example ) of the following form:

(10)Hom 𝒟(L(),)Hom 𝒞(,R()). Hom_{\mathcal{D}}(L(-),-) \;\simeq\; Hom_{\mathcal{C}}(-,R(-)) \,.

This means that for all objects c𝒞c \in \mathcal{C} and d𝒟d \in \mathcal{D} there is a bijection of hom-sets

Hom 𝒟(L(c),d) Hom 𝒞(c,R(d)) (L(c)fd) (cf˜R(d)) \array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ ( L(c) \overset{f}{\to} d ) &\mapsto& (c \overset{\widetilde f}{\to} R(d)) }

which is natural in cc and dd. This isomorphism is called the adjunction hom-isomorphism and the image f˜\widetilde f of a morphism ff under this bijections is called the adjunct of ff. Conversely, ff is called the adjunct of f˜\widetilde f.

Naturality here means that for every pair of morphisms g:c 2c 1g \colon c_2 \to c_1 in 𝒞\mathcal{C} and h:d 1d 2h\colon d_1\to d_2 in 𝒟\mathcal{D}, the resulting square

(11)Hom 𝒟(L(c 1),d 1) ()˜ Hom 𝒞(c 1,R(d 1)) Hom 𝒟(L(g),h) Hom 𝒞(g,R(h)) Hom 𝒟(L(c 2),d 2) ()˜ Hom 𝒞(c 2,R(d 2)) \array{ Hom_{\mathcal{D}}(L(c_1), d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1, R(d_1)) \\ {}^{\mathllap{Hom_{\mathcal{D}}(L(g), h)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(g, R(h))}} \\ Hom_{\mathcal{D}}(L(c_2),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d_2)) }

commutes (Def. ), where the vertical morphisms are given by the hom-functor (Example ).

Explicitly, this commutativity, in turn, means that for every morphism f:L(c 1)d 1f \;\colon\; L(c_1) \to d_1 with adjunct f˜:c 1R(d 1)\widetilde f \;\colon\; c_1 \to R(d_1), the adjunct of the composition is

L(c 1) f d 1 L(g) h L(c 2) d 2˜=c 1 f˜ R(d 1) g R(h) c 2 R(d 2) \widetilde{ \array{ L(c_1) &\overset{f}{\longrightarrow}& d_1 \\ {}^{\mathllap{L(g)}}\big\uparrow && \big\downarrow^{\mathrlap{h}} \\ L(c_2) && d_2 } } \;\;\;=\;\;\; \array{ c_1 &\overset{\widetilde f}{\longrightarrow}& R(d_1) \\ {}^{\mathllap{g}}\big\uparrow && \big\downarrow^{\mathrlap{R(h)}} \\ c_2 && R(d_2) }
Definition

(adjunction unit and counit)

Given a pair of adjoint functors

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}

according to Def. , one says that

  1. for any c𝒞c \in \mathcal{C} the adjunct of the identity morphism on L(c)L(c) is the unit morphism of the adjunction at that object, denoted

    η cid L(c)˜:cR(L(c)) \eta_c \coloneqq \widetilde{id_{L(c)}} \;\colon\; c \longrightarrow R(L(c))
  2. for any d𝒟d \in \mathcal{D} the adjunct of the identity morphism on R(d)R(d) is the counit morphism of the adjunction at that object, denoted

    ϵ d:L(R(d))d \epsilon_d \;\colon\; L(R(d)) \longrightarrow d
Remark

(adjoint triples)

It happens that there are sequences of adjoint functors:

If two functors are adjoint to each other as in Def. , we also say that we have an adjoint pair:

LR. L \;\dashv\; R \,.

It may happen that one functor CC participates on the right and on the left of two such adjoint pairs LCL \,\dashv\, C and CRC \,\dashv\, R (not the same “LL” and “RR” as before!) in which case one may speak of an adjoint triple:

(12)LCR. L \;\dashv\; C \;\dashv\; R \,.

Below in Example we identify adjoint triples as adjunctions of adjunctions.

Similarly there are adjoint quadruples, etc.

Notice that in the case of an adjoint triple (12), the adjunction unit of CRC \dashv R and the adjunction counit of LCL \dashv C (Def. ) provide, for each object XX in the domain of CC, a diagram

(13)L(C(X))AAϵ XAAXAAη XAAR(C(X)) L\big(C(X)\big) \overset { \phantom{AA} \epsilon_X \phantom{AA} } { \longrightarrow } X \overset { \phantom{AA} \eta_X \phantom{AA} } { \longrightarrow } R\big(C(X)\big)

which is usefully thought of as exhibiting the nature of XX as being in between two opposite extreme aspects L(C(X))L\big(C(X)\big) and R(C(X))R\big(C(X)\big) of XX. This is illustrated by the following examples, and formalized by the concept of modalities that we turn to in Def. below.

Example

(floor and ceiling as adjoint functors)

Consider the canonical inclusion

AAιAA \mathbb{Z}_{\leq} \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow} \mathbb{R}_{\leq}

of the integers into the real numbers, both regarded as preorders in the standard way (“lower or equal”). Regarded as full subcategory-inclusion (Def. ) of the corresponding thin categories, via Example , this inclusion functor has both a left and right adjoint functor (Def. ):

forming an adjoint triple (Def. )

(14)()ι(). \lceil(-)\rceil \;\;\dashv\;\; \iota \;\;\dashv\;\; \lfloor (-) \rfloor \,.

The adjunction unit and adjunction counit express that each real number is in between its “opposite extreme integer aspects” (13) given by floor and ceiling

ιxϵ Xxη xιx. \iota \lfloor x \rfloor \;\overset{\epsilon_X}{\leq}\; x \;\overset{\eta_x}{\leq}\; \iota \lceil x \rceil \,.
Proof

First of all, observe that we indeed have functors (Def. )

(),(): \lfloor(-)\rfloor \;,\; \lceil(-)\rceil \;\;\colon\; \mathbb{R} \longrightarrow \mathbb{Z}

since floor and ceiling preserve the ordering relation.

Now in view of the identification of preorders with thin categories in Example , the hom-isomorphism (10) defining adjoint functors of the form ι()\iota \dashv \lfloor(-)\rfloor says for all nn \in \mathbb{Z} and xx \in \mathbb{R}, that we have

nxnx. \underset { \in \mathbb{Z}} {\underbrace{n \leq \lfloor x \rfloor}} \;\;\;\Leftrightarrow\;\;\; \underset { \in \mathbb{R}} {\underbrace{n \leq x }} \,.

This is clearly already the defining condition on the floor function x\lfloor x \rfloor.

Similarly, the hom-isomorphism defining adjoint functors of the form ()ι\lceil(-)\rceil \dashv \iota says that for all nn \in \mathbb{Z} and xx \in \mathbb{R}, we have

xnxn. \underset { \in \mathbb{Z}} {\underbrace{\lceil x \rceil \leq n}} \;\;\;\Leftrightarrow\;\;\; \underset { \in \mathbb{R}} {\underbrace{x \leq n }} \,.

This is evidently already the defining condition on the floor function x\lfloor x \rfloor.

Notice that in both cases the condition of a natural isomorphism in both variables, as required for an adjunction, is automatically satisfied: For let xxx \leq x' and nnn' \leq n, then naturality as in (11) means, again in view of the identifications in Example , that

(nx) (nx) (nx) (nx) \array{ (n \leq \lfloor x \rfloor) &\Leftrightarrow& (n \leq x) \\ \Downarrow && \Downarrow \\ (n' \leq \lfloor x' \rfloor) &\Leftrightarrow& (n' \leq x') \\ \\ \in \mathbb{Z} && \in \mathbb{R} }

Here the logical implications are equivalently functions between sets that are either empty or singletons. But Functions between such sets are unique, when they exist.

Example

(discrete and codiscrete topological spaces)

Consider the “forgetful functorTopUSetTop \overset{U}{\longrightarrow} Set from the category Top of topological spaces (Example ) to the category of sets (Def. ) which sends every topological space to its underlying set.

This has

These hence form an adjoint triple (Remark )

DiscUcoDisc. Disc \;\dashv\; U \;\dashv\; coDisc \,.

Hence the adjunction counit of DiscUDisc \dashv U and the adjunction unit of UcoDiscU \dashv coDisc exhibit every topology on a given set as “in between the opposite extremes” (13) of the discrete and the co-discrete

Disc(U(X))ϵXηcoDisc(U(X)). Disc(U(X)) \overset{\epsilon}{\longrightarrow} X \overset{\eta}{\longrightarrow} coDisc(U(X)) \,.

Lemma

(pre/post-composition with (co-)unit followed by adjunct is adjoint functor)
If a functor CC is the right adjoint

LC:𝒞AALAA AACAA𝒟 L \dashv C \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} L \phantom{AA} }{\longleftarrow} \\ \underset{\phantom{AA} C \phantom{AA} }{\longrightarrow} } \mathcal{D}

in a pair of adjoint functors (Def. ), then its application to any morphism XfY𝒞X \overset{f}{\to} Y \;\;\in \mathcal{C} is equal to the joint operation of pre-composition with the (LC)(L \dashv C)-adjunction counit ϵ X \epsilon^\flat_{X} (Def. ), followed by passing to the (LC)(L \dashv C)-adjunct:

C X,Y=()ϵ X ˜. C_{X, Y} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{X} } \,.

Dually, if CC is a left adjoint

CR:𝒞AACAA AARAA𝒟 C \dashv R \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} C \phantom{AA} }{\longrightarrow} \\ \underset{\phantom{AA} R \phantom{AA} }{\longleftarrow} } \mathcal{D}

then its action on any morphism XfY𝒞X \overset{f}{\to} Y \;\;\in \mathcal{C} equals the joint operation of post-composition with the (CR)(C \dashv R)-adjunction unit η Y \eta^{ \sharp }_{Y} (Def. ), followed by passing to the (CR)(C \dashv R)-adjunct:

η Y ()˜=C X,Y. \widetilde{\eta^\sharp_{Y} \circ (-)} \;=\; C_{X, Y} \,.

In particular, if CC is the middle functor in an adjoint triple (Remark )

LCR:𝒞AALAA AAACAAA AARAA𝒟 L \dashv C \dashv R \;\;\colon\;\; \mathcal{C} \;\; \array{ \overset{\phantom{AA} L \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} C \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} R \phantom{AA} }{\longleftarrow} } \;\; \mathcal{D}

then these two operations coincide:

(15)η Y ()˜=C X,Y=()ϵ X ˜. \widetilde{\eta^\sharp_{Y} \circ (-)} \;=\; C_{X, Y} \;=\; \widetilde{ (-) \circ \epsilon^\flat_{X} } \,.

Proof

For the first equality, consider the following naturality square (4) for the adjunction hom-isomorphism (10):

Hom 𝒟(C(X),C(X)) ()˜ Hom 𝒞(LC(X),X) Hom 𝒟(C(id X),C(f)) Hom X(LC(id X),f) Hom 𝒟(C(X),C(Y)) ()˜ Hom 𝒞(LC(X),Y)AAAAA{CXid CXCX} {LCXϵ X X} {CXC(f)C(Y)} {LCXfϵ X Y} \array{ Hom_{\mathcal{D}}\big( C (X) ,\, C (X) \big) &\overset{\widetilde { (-) }}{\longrightarrow}& Hom_{\mathcal{C}}\big( L C (X) ,\, X \big) \\ {}^{\mathllap{ Hom_{\mathcal{D}}\big( C (id_X) ,\, C(f) \big) }} \big\downarrow && \big\downarrow {}^{ \mathrlap{ Hom_{X}\big( L C (id_X) ,\, f \big) } } \\ Hom_{\mathcal{D}}\big( C (X) ,\, C (Y) \big) &\overset{ \widetilde{ (-) } }{\longleftarrow}& Hom_{\mathcal{C} }( L C (X) ,\, Y ) } \phantom{AAAAA} \array{ \{ C X \overset{id_{C X}}{\to} C X \} &\longrightarrow& \{ L C X \overset{\epsilon^{\flat}_X}{\to} X \} \\ \big\downarrow && \big\downarrow \\ \{ C X \overset{C(f)}{\to} C(Y) \} &\longleftarrow& \{ L C X \overset{f\circ \epsilon^\flat_{X} }{\longrightarrow} Y\} }

Chasing the identity morphism id CYid_{C Y} through this diagram yields the claimed equality, as shown on the right. Here we use that the left adjunct? of the identity morphism is the adjunction counit, as shown.

The second equality is fomally dual:

Hom 𝒟(CY,CY) ()˜ Hom 𝒞(Y,RCY) Hom 𝒟(C(f),C(id Y)) Hom 𝒞(f,RC(id Y)) Hom 𝒟(C(X),C(Y)) ()˜ Hom 𝒞(X,RC(Y))AAAAA{CYid CYCY} {Yη Y RCY} {CXC(f)CY} {Xη Y fRCY} \array{ Hom_{\mathcal{D}} \big( C Y ,\, C Y \big) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{\mathcal{C}} \big( Y ,\, R C Y \big) \\ {}^{\mathllap{ Hom_{\mathcal{D}}\big(C(f), C(id_Y)\big) }} \big\downarrow && \!\!\!\!\! \big\downarrow {}^{\mathrlap{ Hom_{\mathcal{C}}\big( f, R C (id_Y) \big) }} \\ Hom_{\mathcal{D}}\big( C(X),\, C(Y) \big) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{\mathcal{C}}\big( X, R C (Y) \big) } \phantom{AAAAA} \array{ \{ C Y \overset{id_{C Y}}{\to} C Y\} &\longrightarrow& \{ Y \overset{\eta^\sharp_{Y}}{\to} R C Y \} \\ \big\downarrow && \big\downarrow \\ \{ C X \overset{C(f)}{\to} C Y \} &\longleftarrow& \{ X \overset{\eta^\sharp_{Y} \circ f}{\longrightarrow} R C Y \} }



We now turn to a sequence of equivalent reformulations of the condition of adjointness.

Proposition

(general adjuncts in terms of unit/counit)

Consider a pair of adjoint functors

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{\bot} \mathcal{C}

according to Def. , with adjunction units η c\eta_c and adjunction counits ϵ d\epsilon_d according to Def. .

Then

  1. The adjunct f˜\widetilde f of any morphism L(c)fdL(c) \overset{f}{\to} d is obtained from RR and η c\eta_c as the composite

    (16)f˜:cη cR(L(c))R(f)R(d) \widetilde f \;\colon\; c \overset{\eta_c}{\longrightarrow} R(L(c)) \overset{R(f)}{\longrightarrow} R(d)

    Conversely, the adjunct ff of any morphism cf˜R(d)c \overset{\widetilde f}{\longrightarrow} R(d) is obtained from LL and ϵ d\epsilon_d as

    (17)f:L(c)L(f˜)R(L(d))ϵ dd f \;\colon\; L(c) \overset{L(\widetilde f)}{\longrightarrow} R(L(d)) \overset{\epsilon_d}{\longrightarrow} d
  2. The adjunction units η c\eta_c and adjunction counits ϵ d\epsilon_d are components of natural transformations of the form

    η:Id 𝒞RL \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L

    and

    ϵ:LRId 𝒟 \epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
  3. The adjunction unit and adjunction counit satisfy the triangle identities, saying that

    (18)id L(c):L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)

    and

    id R(d):R(d)η R(d)R(L(R(d)))R(ϵ d)R(d) id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)
Proof

For the first statement, consider the naturality square (11) in the form

id L(c) Hom 𝒟(L(c),L(c)) ()˜ Hom 𝒞(c,R(L(c))) Hom 𝒟(L(id),f) Hom 𝒞(id,R(f)) Hom 𝒟(L(c),d) ()˜ Hom 𝒞(c,R(d)) \array{ id_{L(c)} \in & Hom_{\mathcal{D}}(L(c), L(c)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c, R(L(c))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(id), f)}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(id, R(f))}} \\ & Hom_{\mathcal{D}}(L(c), d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}( c, R(d) ) }

and consider the element id L(c 1)id_{L(c_1)} in the top left entry. Its image under going down and then right in the diagram is f˜\widetilde f, by Def. . On the other hand, its image under going right and then down is R(f)η c R(f)\circ \eta_{c}, by Def. . Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown, for the adjunct of ff.

The converse formula follows analogously.

The third statement follows directly from this by applying these formulas for the adjuncts twice and using that the result must be the original morphism:

id L(c) =id L(c)˜˜ =cη cR(L(c))˜ =L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) \begin{aligned} id_{L(c)} & = \widetilde \widetilde { id_{L(c)} } \\ & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) } \\ & = L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c) \end{aligned}

For the second statement, we have to show that for every moprhism f:c 1c 2f \colon c_1 \to c_2 the following square commutes:

c 1 f c 2 η c 1 η c 2 R(L(c 1)) R(L(f)) R(L(c 2)) \array{ c_1 &\overset{f}{\longrightarrow}& c_2 \\ {}^{\mathllap{\eta_{c_1}}}\big\downarrow && \big\downarrow^{\mathrlap{\eta_{c_2}}} \\ R(L(c_1)) &\underset{ R(L(f)) }{\longrightarrow}& R(L(c_2)) }

To see this, consider the naturality square (11) in the form

id L(c 2) Hom 𝒟(L(c 2),L(c 2)) ()˜ Hom 𝒞(c 2,R(L(c 2))) Hom 𝒟(L(f),id L(c 2)) Hom 𝒞(f,R(id L(c 2))) Hom 𝒟(L(c 1),L(c 2)) ()˜ Hom 𝒞(c 1,R(L(c 1))) \array{ id_{L(c_2)} \in & Hom_{\mathcal{D}}(L(c_2), L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2, R(L(c_2))) \\ & {}^{\mathllap{Hom_{\mathcal{D}}(L(f),id_{L(c_2)})}}\big\downarrow && \big\downarrow^{\mathrlap{Hom_{\mathcal{C}}(f, R(id_{L(c_2)}))}} \\ & Hom_{\mathcal{D}}(L(c_1),L(c_2)) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(L(c_1))) }

The image of the element id L(c 2)id_{L(c_2)} in the top left along the right and down is fη c 2 f \circ \eta_{c_2}, by Def. , while its image down and then to the right is L(f)˜=R(L(f))η c 1\widetilde {L(f)} = R(L(f)) \circ \eta_{c_1}, by the previous statement. Commutativity of the diagram means that these two morphisms agree, which is the statement to be shown.

The argument for the naturality of ϵ\epsilon is directly analogous.

Proposition

(adjoint functors equivalent to adjunction in Cat)

Two functors

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{} \mathcal{C}

are an adjoint pair in the sense that there is a natural isomorphism (10) according to Def. , precisely if they participate in an adjunction in the 2-category Cat, meaning that

  1. there exist natural transformations

    η:Id 𝒞RL \eta \;\colon\; Id_{\mathcal{C}} \Rightarrow R \circ L

    and

    ϵ:LRId 𝒟 \epsilon \;\colon\; L \circ R \Rightarrow Id_{\mathcal{D}}
  2. which satisfy the triangle identities

    id L(c):L(c)L(η c)L(R(L(c)))ϵ L(c)L(c) id_{L(c)} \;\colon\; L(c) \overset{L(\eta_c)}{\longrightarrow} L(R(L(c))) \overset{\epsilon_{L(c)}}{\longrightarrow} L(c)

    and

    id R(d):R(d)η R(d)R(L(R(d)))R(ϵ d)R(d) id_{R(d)} \;\colon\; R(d) \overset{\eta_{R(d)}}{\longrightarrow} R(L(R(d))) \overset{R(\epsilon_d)}{\longrightarrow} R(d)
Proof

That a hom-isomorphism (10) implies units/counits satisfying the triangle identities is the statement of the second two items of Prop. .

Hence it remains to show the converse. But the argument is along the same lines as the proof of Prop. : We now define forming of adjuncts by the formula (16). That the resulting assignment ff˜f \mapsto \widetilde f is an isomorphism follows from the computation

f˜˜ =cη cR(L(c))R(f)R(d)˜ =L(c)L(η c)L(R(L(c)))L(R(f))L(R(d))ϵ dd =L(c)L(η c)L(R(L(c)))ϵ L(c)L(c)fd =L(c)fd \begin{aligned} \widetilde {\widetilde f} & = \widetilde{ c \overset{\eta_c}{\to} R(L(c)) \overset{R(f)}{\to} R(d) } \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{L(R(f))}{\to} L(R(d)) \overset{\epsilon_d}{\to} d \\ & = L(c) \overset{L(\eta_c)}{\to} L(R(L(c))) \overset{ \epsilon_{L(c)} }{\to} L(c) \overset{f}{\longrightarrow} d \\ & = L(c) \overset{f}{\longrightarrow} d \end{aligned}

where, after expanding out the definition, we used naturality of ϵ\epsilon and then the triangle identity.

Finally, that this construction satisfies the naturality condition (11) follows from the functoriality of the functors involved, and the naturality of the unit/counit:

c 2 η c 2 R(L(c 2)) g R(L(g)) R(L(g)f) c 1 η c 1 R(L(c 1)) R(f) R(d 1) R(hf) R(h) R(d 2) \array{ c_2 &\overset{ \eta_{c_2} }{\longrightarrow}& R(L(c_2)) \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{R(L(g))}} & \searrow^{\mathrlap{ R( L(g) \circ f ) }} \\ c_1 &\overset{\eta_{c_1}}{\longrightarrow}& R(L(c_1)) &\overset{R(f)}{\longrightarrow}& R(d_1) \\ && & {}_{R( h\circ f)}\searrow & \downarrow^{\mathrlap{ R(h) }} \\ && && R(d_2) }

The condition (10) on adjoint functors LRL \dashv R in Def. implies in particular that for every object d𝒟d \in \mathcal{D} the functor Hom 𝒟(L(),d)Hom_{\mathcal{D}}(L(-),d) is a representable functor with representing object R(d)R(d). The following Prop. observes that the existence of such representing objects for all dd is, in fact, already sufficient to imply that there is a right adjoint functor.

This equivalent perspective on adjoint functors makes manifest that adjoint functors are, if they exist, unique up to natural isomorphism, this is Prop. below.

Proposition

(adjoint functor from objectwise representing objects)

A functor L:𝒞𝒟L \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} has a right adjoint R:𝒟𝒞R \;\colon\; \mathcal{D} \to \mathcal{C}, according to Def. , already if for all objects d𝒟d \in \mathcal{D} there is an object R(d)𝒞R(d) \in \mathcal{C} such that there is a natural isomorphism

Hom 𝒟(L(),d)()˜Hom 𝒞(,R(d)), Hom_{\mathcal{D}}(L(-),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(-,R(d)) \,,

hence for each object c𝒞c \in \mathcal{C} a bijection

Hom 𝒟(L(c),d)()˜Hom 𝒞(c,R(d)) Hom_{\mathcal{D}}(L(c),d) \underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow} Hom_{\mathcal{C}}(c,R(d))

such that for each morphism g:c 2c 1g \;\colon\; c_2 \to c_1, the following diagram commutes

(19)Hom 𝒟(L(c 1),d) ()˜ Hom 𝒞(c 1,R(d)) Hom 𝒞(L(g),id d) Hom 𝒞(f,id R(d)) Hom 𝒟(L(c 2),d) ()˜ Hom 𝒞(c 2,R(d)) \array{ Hom_{\mathcal{D}}(L(c_1),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_1,R(d)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}(L(g),id_d) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, id_{R(d)} ) }} \\ Hom_{\mathcal{D}}(L(c_2),d) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(c_2,R(d)) }

(This is as in (11), except that only naturality in the first variable is required.)

In this case there is a unique way to extend RR from a function on objects to a function on morphisms such as to make it a functor R:𝒟𝒞R \colon \mathcal{D} \to \mathcal{C} which is right adjoint to LL. , and hence the statement is that with this, naturality in the second variable is already implied.

Proof

Notice that

  1. in the language of presheaves (Example ) the assumption is that for each d𝒟d \in \mathcal{D} the presheaf

    Hom 𝒟(L(),d)[𝒟 op,Set] Hom_{\mathcal{D}}(L(-),d) \;\in\; [\mathcal{D}^{op}, Set]

    is represented (7) by the object R(d)R(d), and naturally so.

  2. In terms of the Yoneda embedding (Prop. )

    y:𝒟[𝒟 op,Set] y \;\colon\; \mathcal{D} \hookrightarrow [\mathcal{D}^{op}, Set]

    we have

    (20)Hom 𝒞(,R(d))=y(R(d)) Hom_{\mathcal{C}}(-,R(d)) = y(R(d))

The condition (11) says equivalently that RR has to be such that for all morphisms h:d 1d 2h \;\colon\; d_1 \to d_2 the following diagram in the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] commutes

Hom 𝒟(L(),d 1) ()˜ Hom 𝒞(,R(d 1)) Hom 𝒞(L(),h) Hom 𝒞(,R(h)) Hom 𝒟(L(),d 2) ()˜ Hom 𝒞(,R(d 2)) \array{ Hom_{\mathcal{D}}(L(-),d_1) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-,R(d_1)) \\ {}^{\mathllap{ Hom_{\mathcal{C}}( L(-) , h ) }} \big\downarrow && \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( -, R(h) ) }} \\ Hom_{\mathcal{D}}(L(-),d_2) &\underoverset{\simeq}{\widetilde{(-)}}{\longrightarrow}& Hom_{\mathcal{C}}(-, R(d_2)) }

This manifestly has a unique solution

y(R(h))=Hom 𝒞(,R(h)) y(R(h)) \;=\; Hom_{\mathcal{C}}(-,R(h))

for every morphism h:d 1d 2h \colon d_1 \to d_2 under y(R())y(R(-)) (20). But the Yoneda embedding yy is a fully faithful functor (Prop. ), which means that thereby also R(h)R(h) is uniquely fixed.

We consider one more equivalent characterization of adjunctions:

Definition

(universal morphism)

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be two categories (Def. ) and let R:𝒟𝒞R \;\colon\; \mathcal{D} \to \mathcal{C} be a functor (Def. )

Then for c𝒞c\in \mathcal{C} an object, a universal morphism from cc to RR is

  1. an object L(c)𝒟L(c)\in \mathcal{D},

  2. a morphism η c:cR(L(c))\eta_c \;\colon\; c \to R(L(c)), to be called the unit,

such that for any d𝒟d\in \mathcal{D}, any morphism f:cR(d)f \colon c\to R(d) factors through this unit η c\eta_c as

(21)f=R(f˜)η cAAAA c η c f R(L(c)) R(f˜) R(d) L(c) f˜ d f \;=\; R(\widetilde f) \circ \eta_c \phantom{AAAA} \array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \widetilde f}{\longrightarrow}&& d }

for a unique morphism f˜:L(c)d\widetilde f \;\colon\; L(c) \longrightarrow d, to be called the adjunct of ff.

Proposition

(collection of universal morphisms equivalent to adjoint functor)

Let R:𝒟𝒞R \;\colon\; \mathcal{D} \to \mathcal{C} be a functor (Def. ). Then the following are equivalent:

  1. RR has a left adjoint functor L:𝒞𝒟L \colon \mathcal{C} \to \mathcal{D} according to Def. .

  2. For every object c𝒞c \in \mathcal{C} there is a universal morphism cη cR(L(c))c \overset{\eta_c}{\longrightarrow} R(L(c)), according to Def. .

Proof

In one direction, assume a left adjoint LL is given. Define the would-be universal arrow at c𝒞c \in \mathcal{C} to be the unit of the adjunction η c\eta_c via Def. . Then the statement that this really is a universal arrow is implied by Prop. .

In the other direction, assume that universal arrows η c\eta_c are given. The uniqueness clause in Def. immediately implies bijections

Hom 𝒟(L(c),d) Hom 𝒞(c,R(d)) (L(c)f˜d) (cη cR(L(c))R(f˜)R(d)) \array{ Hom_{\mathcal{D}}(L(c),d) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(c,R(d)) \\ \left( L(c) \overset{\widetilde f}{\to} d \right) &\mapsto& \left( c \overset{\eta_c}{\to} R(L(c)) \overset{ R(\widetilde f) }{\to} R(d) \right) }

Hence to satisfy (10) it remains to show that these are natural in both variables. In fact, by Prop. it is sufficient to show naturality in the variable dd. But this is immediate from the functoriality of RR applied in (21): For h:d 1d 2h \colon d_1 \to d_2 any morphism, we have

c η c f R(L(c)) R(f˜) R(d 1) R(hf˜) R(h) R(d 2) \array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R (L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d_1) \\ && {}_{\mathllap{ R( h\circ \widetilde f ) }}\searrow && \downarrow^{\mathrlap{R(h)}} \\ && && R(d_2) }

The following equivalent formulation (Prop. ) of universal morphisms is often useful:

Example

(comma category)

Let 𝒞\mathcal{C} be a category, let c𝒞c \in \mathcal{C} be any object, and let F:𝒟𝒞F \;\colon\; \mathcal{D} \to \mathcal{C} be a functor.

  1. The comma category c/Fc/F is the category whose objects are pairs consisting of an object d𝒟d \in \mathcal{D} and morphisms XfF(d)X \overset{f}{\to} F(d) in 𝒞\mathcal{C}, and whose morphisms (d 1,X 1,f 1)(d 2,X 2,f 2)(d_1,X_1,f_1) \to (d_2,X_2,f_2) are the morphisms X 1gX 2X_1 \overset{g}{\longrightarrow} X_2 in 𝒞\mathcal{C} that make a commuting triangle (Def. ):

    f 2F(g)=f 1AAAAAAX 1 AAgAA X 2 F(X 1) AAF(g)AA F(X 2) f 1 f 2 c f_2\circ F(g) \;=\; f_1 \phantom{AAAAAA} \array{ X_1 && \overset{\phantom{AA} g \phantom{AA}}{\longrightarrow} && X_2 \\ F(X_1) && \overset{\phantom{AA} F(g) \phantom{AA}}{\longrightarrow} && F(X_2) \\ & {}_{\mathllap{f_1}}\searrow && \swarrow_{\mathrlap{f_2}} \\ && c }

    There is a canonical functor

    F/c 𝒟. \array{ F/c &\overset{}{\longrightarrow}& \mathcal{D} } \,.
  2. The comma category F/cF/c is the category whose objects are pairs consisting of an object d𝒟d \in \mathcal{D} and a morphism F(d)fXF(d) \overset{f}{\to} X in 𝒞\mathcal{C}, and whose morphisms (d 1,X 1,f 1)(d 2,X 2,f 2)(d_1,X_1,f_1) \to (d_2,X_2,f_2) are the morphisms X 1gX 2X_1 \overset{g}{\longrightarrow} X_2 in 𝒞\mathcal{C} that make a commuting triangle (Def. ):

    f 2F(g)=f 1AAAAAA c f 1 f 2 F(X 1) AAF(g)AA F(X 2) X 1 AAgAA X 2 f_2\circ F(g) \;=\; f_1 \phantom{AAAAAA} \array{ && c \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} \\ F(X_1) && \underset{\phantom{AA} F(g) \phantom{AA}}{\longrightarrow} && F(X_2) \\ X_1 && \underset{ \phantom{AA} g \phantom{AA} }{\longrightarrow} && X_2 }

    Again, there is a canonical functor

    (22)c/F 𝒟] \array{ c/F &\overset{}{\longrightarrow}& \mathcal{D} } ]

With this definition, the following is evident:

Proposition

(universal morphisms are initial objects in the comma category)

Let 𝒞R𝒟\mathcal{C} \overset{R}{\longrightarrow} \mathcal{D} be a functor and d𝒟d \in \mathcal{D} an object. Then the following are equivalent:

  1. dη dR(c)d \overset{\eta_d}{\to} R(c) is a universal morphism into R(c)R(c) (Def. );

  2. (d,η d)(d, \eta_d) is the initial object (Def. ) in the comma category d/Rd/R (Example ).

\,

\,

After these equivalent characterizations of adjoint functors, we now consider some of their main properties:

Proposition

(adjoint functors are unique up to natural isomorphism)

The left adjoint or right adjoint to a functor (Def. ), if it exists, is unique up to natural isomorphism (Def. ).

Proof

Suppose the functor L:𝒟𝒞L \colon \mathcal{D} \to \mathcal{C} is given, and we are asking for uniqueness of its right adjoint, if it exists. The other case is directly analogous.

Suppose that R 1,R 2:𝒞𝒟R_1, R_2 \;\colon\; \mathcal{C} \to \mathcal{D} are two functors which both are right adjoint to LL. Then for each d𝒟d \in \mathcal{D} the corresponding two hom-isomorphisms (10) combine to say that there is a natural isomorphism/

Φ d:Hom 𝒞(,R 1(d))Hom 𝒞(,R 2(d)) \Phi_d \;\colon\; Hom_{\mathcal{C}}(-,R_1(d)) \;\simeq\; Hom_{\mathcal{C}}(-,R_2(d))

As in the proof of Prop. , the Yoneda lemma implies that

Φ d=y(ϕ d) \Phi_d \;=\; y( \phi_d )

for some isomorphism

ϕ d:R 1(d)R 2(d). \phi_d \;\colon\; R_1(d) \overset{\simeq}{\to} R_2(d) \,.

But then the uniqueness statement of Prop. implies that the collection of these isomorphisms for each object constitues a natural isomorphism between the functors (Def. ).

Proposition

(characterization of epi/mono/iso (co-)unit of adjunction)

Let

LR:𝒟ARAALA𝒞 L \dashv R \;\colon\; \mathcal{D} \underoverset {\underset{\phantom{A}R\phantom{A}}{\longrightarrow}} {\overset{\phantom{A}L\phantom{A}}{\longleftarrow}} {\bot} \mathcal{C}

be a pair of adjoint functors (Def. ).

Recall the definition of

  1. adjunction unit/counit, from Def. )

  2. faithful/fully faithful functor from Def.

  3. mono/epi/isomorphism from Def. and Def. .

The following holds:

Proof

This follows directly by Lemma , using the definition of epi/monomorphism (Def. ) and the characterization of isomorphism from Example .

To complete this pattern, we will see below in Prop. that following are equivalent:

Proposition

(right/left adjoint functors preserve monomorphism/epimorphisms and terminal/initial objects)

Every right adjoint functor (Def. ) preserves

  1. terminal objects (Def. ),

  2. monomorphisms (Def. )

Every left adjoint functor (Def. ) preserves

  1. initial objects (Def. ),

  2. epimorphisms (Def. ).

Proof

This is immediate from the adjunction hom-isomorphism (10), but we spell it out:

We consider the first case, the second is formally dual (Example ). So let R:𝒞𝒟R \;\colon\; \mathcal{C} \to \mathcal{D} be a right adjoint functor with left adjoint LL.

Let *𝒞\ast \in \mathcal{C} be a terminal object (Def. ). We need to show that for every object d𝒟d \in \mathcal{D} the hom-set Hom 𝒟(d,R(*))*Hom_{\mathcal{D}}(d,R(\ast)) \simeq \ast is a singleton. But by the hom-isomorphism (10) we have a bijection

Hom 𝒹(d,R(*)) Hom 𝒞(L(d),*) *, \begin{aligned} Hom_{\mathcal{d}}(d,R(\ast)) & \simeq Hom_{\mathcal{C}}(L(d), \ast) \\ & \simeq \ast \,, \end{aligned}

where in the last step we used that *\ast is a terminal object, by assumption.

Next let c 1fc 2c_1 \overset{f}{\hookrightarrow} c_2 be a monomorphism. We need to show that for d𝒟d \in \mathcal{D} any object, the hom-functor out of dd yields a monomorphism

Hom 𝒟(d,R(f)):Hom 𝒟(d,R(c 1))Hom 𝒟(d,R(c 2)). Hom_{\mathcal{D}}(d, R(f)) \;\colon\; Hom_{\mathcal{D}}(d, R(c_1)) \hookrightarrow Hom_{\mathcal{D}}(d, R(c_2)) \,.

Now consider the following naturality square (11) of the adjunction hom-isomorphism (10):

Hom 𝒟(d,R(c 1)) Hom 𝒞(L(d),c 1) Hom 𝒟(d,R(f)) mono Hom 𝒞(L(d),f) Hom 𝒟(d,R(c 2)) Hom 𝒞(L(d),c 2) \array{ Hom_{\mathcal{D}}(d, R(c_1)) &\simeq& Hom_{\mathcal{C}}(L(d), c_1) \\ {}^{ \mathllap{ Hom_{\mathcal{D}}(d,R(f)) } }\big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{\mathcal{C}}( L(d),f ) } }_{\mathrlap{mono}} \\ Hom_{\mathcal{D}}(d, R(c_2)) &\simeq& Hom_{\mathcal{C}}(L(d), c_2) }

Here the right vertical function is an injective function, by assumption on ff and the definition of monomorphism. Since the two horizontal functions are bijections, this implies that also Hom 𝒹(d,R(f))Hom_{\mathcal{d}}(d,R(f)) is an injection.

But the main preservation property of adjoint functors is that adjoints preserve (co-)limits. This we discuss as Prop. below, after introducing limits and colimits in Def. below.

\,

Prop. says that adjoint functors are equivalenty “adjunctions in Cat”, as defined there. This is a special case of a general more abstract concept of adjunction, that is useful:

Definition

(strict 2-category)

A strict category 𝒞\mathcal{C} is

  1. a class Obj 𝒞Obj_{\mathcal{C}}, called the class of objects;

  2. for each pair X,YObj 𝒞X,Y \in Obj_{\mathcal{C}} of objects, a small category Hom 𝒞(X,Y)CatHom_{\mathcal{C}}(X,Y) \in Cat (Def. ), called the hom-category from XX to YY.

    We denote the objects of this hom-category by arrows like this:

    XfYObj Hom 𝒞(X,Y) X \overset{f}{\longrightarrow} Y \;\;\in Obj_{Hom_{\mathcal{C}}(X,Y)}

    and call them the 1-morphisms of 𝒞\mathcal{C},

    and we denote the morphisms in the hom-category by double arrows, like this:

    X ϕgfY X \underoverset {\underset{g}{\longrightarrow}} {\overset{f}{\longrightarrow}} {\Downarrow{}^{\mathrlap{\phi}}} Y

    and call these the 2-morphisms of 𝒞\mathcal{C};

  3. for each object XObj 𝒞X \in Obj_{\mathcal{C}} a 1-morphism

    Xid XXHom 𝒞(X,X) X \overset{id_X}{\to} X \;\; \in Hom_{\mathcal{C}}(X,X)

    called the identity morphism on XX;

  4. for each triple X 1,X 2,X 3ObjX_1, X_2, X_3 \in Obj of objects, a functor (Def. )

    Hom 𝒞(X 1,X 2) × Hom 𝒞(X 2,X 3) X 1,X 2,X 3 Hom 𝒞(X 1,X 3) X 1fX 2 , X 2fX 3 X 1gfX 3 \array{ Hom_{\mathcal{C}}(X_1, X_2) &\times& Hom_{\mathcal{C}}(X_2, X_3) &\overset{\circ_{X_1,X_2,X_3}}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \\ X_1 \overset{f}{\to} X_2 &,& X_2 \overset{f}{\to} X_3 &\mapsto& X_1 \overset{ g \circ f }{\longrightarrow} X_3 }

    from the product category (Example ) of hom-categories, called composition;

such that:

  1. for all pairs of objects X,YObj 𝒞X,Y \in Obj_{\mathcal{C}} unitality holds:

    the functors of composition with identity morphisms are identity functors

    ()id X=id Hom 𝒞(X,Y)AAAAid Y()=id Hom 𝒞(X,Y) (-) \circ id_X \;=\; id_{ Hom_{\mathcal{C}}(X,Y) } \phantom{AAAA} id_Y \circ (-) \;=\; id_{ Hom_{\mathcal{C}}(X,Y) }
  2. for all quadruples of objects X 1,X 2,X 3,X 4Obj 𝒞X_1, X_2, X_3, X_4 \in Obj_{\mathcal{C}} composition satifies associativity, in that the following two composite functors are equal:

    Hom 𝒞(X 1,X 2)×Hom 𝒞(X 2,X 3)×Hom 𝒞(X 3,X 4) (()())() Hom 𝒞(X 1,X 3)×Hom 𝒞(X 3,X 4) ()(()()) ()() Hom 𝒞(X 1,X 2)×Hom 𝒞(X 2,X 4) ()() Hom 𝒞(X 1,X)4) \array{ Hom_{\mathcal{C}}(X_1, X_2) \times Hom_{\mathcal{C}}(X_2, X_3) \times Hom_{\mathcal{C}}(X_3, X_4) &\overset{((-)\circ (-))\circ (-)}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X_3) \times Hom_{\mathcal{C}}(X_3, X_4) \\ {}^{ \mathllap{ (-) \circ ( (-) \circ (-) ) } }\Big\downarrow && \Big\downarrow{}^{ (-) \circ (-) } \\ Hom_{\mathcal{C}}(X_1, X_2) \times Hom_{\mathcal{C}}(X_2, X_4) &\underset{(-)\circ (-)}{\longrightarrow}& Hom_{\mathcal{C}}(X_1, X)4) }

The archetypical example of a strict 2-category is the category of categories:

Example

(2-category of categories)

There is a strict 2-category (Def. ) Cat whose

with the evident composition operations.

With a concept of 2-category in hand, we may phrase Prop. more abstractly:

Definition

(adjunction in a 2-category)

Let 𝒞\mathcal{C} be a strict 2-category (Def. ). Then an adjunction in 𝒞\mathcal{C} is

  1. a pair of objects 𝒞,𝒟Obj 𝒞\mathcal{C}, \mathcal{D} \in Obj_{\mathcal{C}};

  2. 1-morphisms

    𝒟AARAAL𝒞 \mathcal{D} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {} \mathcal{C}

    called the left adjoint LL and right adjoint RR;

  3. 2-morphisms

    id 𝒞ηRLid_{\mathcal{C}} \overset{\eta}{\Rightarrow} R \circ L, called the adjunction unit

    LRϵid 𝒟L \circ R \overset{\epsilon}{\Rightarrow} id_{\mathcal{D}}, called the adjunction counit

such that the following triangle identities hold:

We denote this situation by

𝒟AARAAL𝒞 \mathcal{D} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{C}

Hence via Example , Prop. says that an adjoint pair of functors is equivalente an adjunction in the general sense of Def. , realized in the very large strict 2-category Cat of categories.

This more abstract perspecive on adjunctions allow us now to understand “duality of dualities” as adjunction in a 2-category of adjunctions:

Example

(strict 2-category of categories with adjoint functors between them)

Let Cat AdjCat_{Adj} be the strict 2-category which is defined just as Cat (Def. ) but with the 1-morphisms being functors that are required to be left adjoints (Def. ).

Since adjoints are unique up to natural isomorphism (Prop. ), this may be thought of as a 2-category whose 1-morphisms are adjoint pairs of functors.

Example

(adjunctions of adjoint pairs are adjoint triples)

An adjunction (Def. ) in the 2-category Cat AdjCat_{Adj} of categories with adjoint functors between them (Example ) is equivalently an adjoint triple of functors (Remark ):

The adjunction says that two left adjoint functors L 1L_1 and L 2L_2, which, hence each participate in an adjoint pair

L 1R 1AAAAL 2R 2 L_1 \dashv R_1 \phantom{AAAA} L_2 \dashv R_2

form themselves an adjoint pair

L 1L 2. L_1 \dashv L_2 \,.

By essentiall uniqueness of adjoints (Prop. ) this implies a natural isomorphism R 1L 2R_1 \simeq L_2 and hence an adjoint triple:

𝒟A aL 1A a A aR 1L 2 A aR 2𝒞 \mathcal{D} \array{ \underoverset{\bot \phantom{\simeq A_a}}{ L_1 \phantom{\simeq A_a} }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot}{R_1 \simeq L_2}{\longrightarrow} \\ \overset{ \phantom{A_a \simeq} R_2 }{\longleftarrow} } \mathcal{C}

Example suggest to consider a slight variant of the concept of strict 2-categories which allows to make the duality between left adjoints and right adjoints explicit:

Definition

(double category)

A double category 𝒞\mathcal{C} is

  1. a pair of categories 𝒞 h\mathcal{C}_h, 𝒞 v\mathcal{C}_v (Def. ) which share the same class of objects: Obj 𝒞 1=Obj 𝒞 2Obj_{\mathcal{C}_1} = Obj_{\mathcal{C}_2}, to be called the class Obj 𝒞Obj_{\mathcal{C}} of objects of 𝒞\mathcal{C}

    where the morphisms of 𝒞 h\mathcal{C}_h are to be called the horizontal morphisms of 𝒞\mathcal{C},

    while the morphisms of 𝒞 v\mathcal{C}_v are to be called the vertical morphisms of 𝒞\mathcal{C},

  2. for each quadruple of objects a,b,c,d,eObj 𝒞a,b,c,d,e \in Obj_{\mathcal{C}} and pairs of pairs of horizontal/vertical morphisms of the form

    a f𝒞 h b h𝒞 v k𝒞 v c g𝒞 h \array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow && \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& }

    a set 2Hom(f,g,h,k)2Hom(f,g,h,k), to be called the set of 2-morphisms of 𝒞\mathcal{C} between the given 1-morphisms, whose elements we denote by

    a f𝒞 h b h𝒞 v k𝒞 v c g𝒞 h d \array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow &\swArrow& \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& d }
  3. a horizontal and a vertical composition operation of 2-morphisms which is unitality and associative in both directions in the evident way, which respects composition in 𝒞 h\mathcal{C}_h and 𝒞 v\mathcal{C}_v, and such that horizontal and vertical composition commute over each other in the evident way.

Example

(double category of squares of a strict 2-category)

Let 𝒞\mathcal{C} be a strict 2-category (Def. ). Then its double category of squares Sq(𝒞)Sq(\mathcal{C}) is the double category (Def. ) whose

  • objects are those of 𝒞\mathcal{C};

  • horizontal morphisms and vertical morphisms are both the 1-morphisms of 𝒞\mathcal{C};

  • 2-morphisms

    a f𝒞 h b h𝒞 v ϕ k𝒞 v c g𝒞 h d \array{ a &\overset{f \in \mathcal{C}_h}{\longrightarrow}& b \\ {}^{\mathllap{h \in \mathcal{C}_v}}\big\downarrow &{}^{\mathllap{\phi}}\swArrow& \big\downarrow{}^{\mathrlap{k \in \mathcal{C}_v}} \\ c &\underset{g \in \mathcal{C}_h}{\longrightarrow}& d }

    are the 2-morphisms of 𝒞\mathcal{C} between the evident composites of 1-morphisms:

    kfϕgh k \circ f \overset{\phi}{\Rightarrow} g\circ h

and composition is given by the evident compositions in 𝒞\mathcal{C}.

Remark

(strict and weak 2-functors)

Given two strict 2-categories (Def. ) or double categories (Def. ), 𝒞,𝒟\mathcal{C}, \mathcal{D}, there is an evident notion of 2-functor or double functor

𝒞F𝒟 \mathcal{C} \overset{F}{\longrightarrow} \mathcal{D}

between them, namely functions on objects, 1-morphisms and 2-morphisms which respect all the composition operations and identity morphisms.

These are also called strict 2-functors.

This is in contrast to a more flexible concept of weak 2-functors, often called pseudofunctors, which respect composition of 1-morphisms only up to invertible 2-morphisms (which themselves are required to satisfy some coherence condition):

Y F(f) ρ F(G) X F(gf) Z \array{ && Y \\ & {}^{\mathllap{ F(f) }}\nearrow &\Downarrow{}^{\rho}_{}\simeq& \searrow^{\mathrlap{F(G)}} \\ X && \underset{F(g \circ f)}{\longrightarrow} && Z }

We will see an important example of a weak double functor in the construction of derived functors of Quillen functors, below in Prop. .

\,

Equivalences

We have seen functors (Def. ) as the homomorphisms between categories (Def. ). But functors themselves are identified only up to natural isomorphism (Def. ), reflective the fact that they are the 1-morphisms in a 2-category of categories (Example ). This means that in identifying two categories, we should not just ask for isomorphisms between them, hence for a functor between them that has a strict inverse morphism, but just for an inverse up to natural isomorphism.

This is called an equivalence of categories (Def. below). A particularly well-behaved equivalence of categories is an equivalence exhibited by an adjoint pair of functors, called an adjoint equivalence of categories (Def. below). In fact every equivalence of categories may be improved to an adjoint equivalence (Prop. ).

\,

Definition

(adjoint equivalence of categories)

Let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two categories (Def. ). Then an adjoint equivalence of categories between them is a pair adjoint functors (Def. )

𝒞A ARL𝒟 \array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{A} \phantom{{}_{\bot}}\simeq_{\bot} \phantom{A}} \mathcal{D} }

such that their unit η\eta and counit ϵ\epsilon (Def. ) are natural isomorphisms (as opposed to just being natural transformations)

η:id 𝒟RLAAAandAAAϵ:LRid 𝒞. \eta\;\colon\; id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} \epsilon\;\colon\; L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.

There is also the following, seemingly weaker, notion:

Definition

(equivalence of categories)

Let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two categories (Def. ). Then an equivalence of categories

𝒞AAAARL𝒟 \array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA} \simeq \phantom{AA}} \mathcal{D} }

is a pair of functors back and forth, as shown (Def. ), together with natural isomorphisms (Def. ) between their composition and the identity functors:

id 𝒟RLAAAandAAALRid 𝒞. id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.

If a functor participates in an equivalence of categories, that functor alone is usually already called an equivalence of categories. If there is any equivalence of categories between two categories, these categories are called equivalent.

Proposition

(every equivalence of categories comes from an adjoint equivalence of categories)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two categories (Def. ). Then the they are equivalent (Def. ) precisely if there exists an adjoint equivalence of categories between them (Def. ).

Moreover, let R:𝒞𝒟R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a functor (Def. ) which participates in an equivalence of categories (Def. ). Then for every functor L:𝒟𝒞L \;\colon\; \mathcal{D} \to \mathcal{C} equipped with a natural isomorphism

η:id 𝒟RL \eta \;\colon\; id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L

there exists a natural isomorphism

ϵ:LRid 𝒞 \epsilon \;\colon\; L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}}

which completes this to an adjoint equivalence of categories (Def. ).

Inside every adjunction sits its maximal adjoint equivalence:

Proposition

(fixed point equivalence of an adjunction)

Let

𝒟AAAARL𝒞 \mathcal{D} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}

be a pair of adjoint functors (Def. ). Say that

  1. an object c𝒞c \in \mathcal{C} is a fixed point of the adjunction if its adjunction unit (Def. ) is an isomorphism (Def. )

    cη cRL(c) c \underoverset{\simeq}{\eta_c}{\longrightarrow} R L (c)

    and write

    𝒞 fix𝒞 \mathcal{C}_{fix} \hookrightarrow \mathcal{C}

    for the full subcategory on these fixed objects (Example )

  2. an object d𝒟d \in \mathcal{D} is a fixed point of the adjunction if its adjunction counit (Def. ) is an isomorphism (Def. )

    LR(d)ϵ d L R(d) \underoverset{\simeq}{\epsilon_d}{\longrightarrow}

    and write

    𝒟 fix𝒟 \mathcal{D}_{fix} \hookrightarrow \mathcal{D}

    for the full subcategory on these fixed objects (Example )

Then the adjunction (co-)restrics to an adjoint equivalence (Def. ) on these full subcategories of fixed points

𝒟 fixA ARL𝒞 fix \mathcal{D}_{fix} \underoverset {\underset{ R }{\longrightarrow}} {\overset{ L }{\longleftarrow}} {\phantom{A}\phantom{{}_{\bot}}\simeq_{\bot}\phantom{A}} \mathcal{C}_{fix}
Proof

It is sufficient to see that the functors (co-)restrict as claimed, for then the restricted adjunction unit/counit are isomorphisms by definition, and hence exhibit an adjoint equivalence.

Hence we need to show that

  1. for c𝒞 fix𝒞c \in \mathcal{C}_{fix} \hookrightarrow \mathcal{C} we have that η R(d)\eta_{R(d)} is an isomorphism;

  2. for d𝒟 fix𝒟d \in \mathcal{D}_{fix} \hookrightarrow \mathcal{D} we have that ϵ L(c)\epsilon_{L(c)} is an isomorphism.

For the first case we claim that R(η d)R(\eta_{d}) provides an inverse: by the triangle identity (18) it is a right inverse, but by assumption it is itself an invertible morphism, which implies that η R(d)\eta_{R(d)} is an isomorphism.

The second claim is formally dual.

\,

Modalities

Generally, a full subcategory-inclusion (Def. ) may be thought of as a consistent proposition about objects in a category: The objects in the full subcategory are those that have the given property.

This basic situation becomes particularly interesting when the inclusion functor has a left adjoint or a right adjoint (Def. ), in which case one speaks of a reflective subcategory, or a coreflective subcategory, respectively (Def. below). The adjunction now implies that each object is reflected or coreflected into the subcategory, and equipped with a comparison morphism to or from its (co-)reflection (the adjunction (co-)unit, Def. ). This comparison morphism turns out to always be an idempotent (co-)projection, in a sense made precise by Prop. below.

This means that, while any object may not fully enjoy the property that defines the subcategory, one may ask for the “aspect” of it that does, which is what is (co-)projected out. Regarding objects only via these aspects of them hence means to regard them only locally (where they exhibit that aspect) or only in the mode of focus on this aspect. Therefore one also calls the (co-)reflection operation into the given subcategory a (co-)localization or (co-)modal operator, or modality, for short (Def. below).

One finds that (co-)modalities are a fully equivalent perspective on the (co-)reflective subcategories of their fully (co-)modal objects (Def. below), this is the statement of Prop. below.

Another alternative perspective on this situation is given by the concept of localization of categories (Def. below), which is about universally forcing a given collection of morphisms (“weak equivalences”, Def. below) to become invertible. A reflective localization is equivalently a reflective subcategory-inclusion (Prop. below), and this exhibits the modal objects (Def. below) as equivalently forming the full subcategory of local objects (Def. below).

Conversely, every reflection onto full subcategories of SS-local objects (Def. below) satisfies the universal property of a localization at SS with respect to left adjoint functors (Prop. below).

In conclusion, we have the following three equivalent perspectives on modalities.

A\phantom{A}reflective subcategoryA\phantom{A}A\phantom{A}modal operatorA\phantom{A}A\phantom{A}reflective localizationA\phantom{A}
A\phantom{A}object in reflectiveA\phantom{A}
A\phantom{A}full subcategoryA\phantom{A}
A\phantom{A}modal objectA\phantom{A}A\phantom{A}local objectA\phantom{A}

\,

Definition

(reflective subcategory and coreflective subcategory)

Let 𝒟\mathcal{D} be a category (Def. ) and

𝒞AAιAA𝒟 \mathcal{C} \overset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow} \mathcal{D}

a full subcategory-inclusion (hence a fully faithful functor Def. ). This is called:

  1. a reflective subcategory inclusion if the inclusion functor ι\iota has a left adjoint LL def. )

    𝒞AAιAAL𝒟, \mathcal{C} \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D} \,,

    then called the reflector;

  2. a coreflective subcategory-inclusion if the inclusion functor ι\iota has a right adjoint RR (def. )

    𝒞RAAιAA𝒟, \mathcal{C} \underoverset \underset{R}{\longleftarrow} {\overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\bot} \mathcal{D} \,,

    then called the coreflector.

Example

(reflective subcategory inclusion of sets into small groupoids)

There is a reflective subcategory-inclusion (Def. )

SetAAAAπ 0Grpd Set \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{\pi_0}{\longleftarrow}} {\bot} Grpd

of the category of sets (Example ) into the category Grpd (Example ) of small groupoids (Example ) where

\,

We now re-consider the concept of reflective subcategories from the point of view of modalities:

Definition

(modality)

Let 𝒟\mathcal{D} be a category (Def. ). Then

  1. a modal operator on 𝒟\mathcal{D} is

    1. an endofunctor

      :𝒟𝒟 \bigcirc \;\colon\; \mathcal{D} \to \mathcal{D}

      whose full essential image we denote by

      Im()AAιAA𝒟, Im(\bigcirc) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} \mathcal{D} \,,
    2. a natural transformation (Def. )

      (23)Xη XX X \overset{\eta_X}{\longrightarrow} \bigcirc X

      for all objects X𝒟X \in \mathcal{D}, to be called the unit morphism;

    such that:

    • for every object YIm()𝒟Y \in Im(\bigcirc) \hookrightarrow \mathcal{D} in the essential image of \bigcirc, every morphism ff into YY factors uniquely through the unit (23)

      X η X f X ! Y Im() \array{ && X \\ & {}^{\mathllap{ \eta_X }}\swarrow && \searrow^{\mathrlap{f}} \\ \mathrlap{\bigcirc X\;\;\;\;} && \underset{\exists !}{\longrightarrow} && Y & \in Im(\bigcirc) }

      which equivalently means that if YIm()Y \in Im(\bigcirc) the operation of precomposition with the unit η X\eta_X yields a bijection of hom-sets

      (24)()η X:Hom 𝒟(X,Y)AAAAHom 𝒟(X,Y), (-)\circ \eta_X \;\colon\; Hom_{\mathcal{D}}(\bigcirc X, Y) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} Hom_{\mathcal{D}}(X, Y) \,,
  2. a comodal operator on 𝒟\mathcal{D} is

    1. an endofunctor

      :𝒟𝒟 \Box \;\colon\; \mathcal{D} \to \mathcal{D}

      whose full essential image we denote by

      Im()AAιAA𝒟 Im( \Box ) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} \mathcal{D}
    2. a natural transformation (Def. )

      (25)Xϵ XX \Box X \overset{ \epsilon_X }{\longrightarrow} X

      for all objects X𝒟X \in \mathcal{D}, to be called the counit morphism;

    such that:

    • for every object YIm()𝒟Y \in Im( \Box ) \hookrightarrow \mathcal{D} in the essential image of \Box, every morphism ff out of YY factors uniquely through the counit (23)

      X ϵ X f X ! YIm() \array{ && X \\ & {}^{\mathllap{\epsilon_X}}\nearrow && \nwarrow^{\mathrlap{f}} \\ \mathrlap{\Box X\;\;\;} && \underset{\exists !}{\longleftarrow} && Y \in Im( \Box ) }

      which equivalently means that if YIm()Y \in Im(\bigcirc) the operation of postcomposition with the counit ϵ X\epsilon_X yields a bijection of hom-sets

      (26)ϵ X():Hom 𝒟(Y,X)AAAAHom 𝒟(Y,X), \epsilon_X \circ (-) \;\colon\; Hom_{\mathcal{D}}(Y, \Box X) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} Hom_{\mathcal{D}}(Y , X) \,,
Proposition

(modal operators equivalent to reflective subcategories)

If

𝒞AAιAAL𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

is a reflective subcategory-inclusion (Def. ). Then the composite

ιL:𝒟𝒟 \bigcirc \;\coloneqq\; \iota \circ L \;\colon\; \mathcal{D} \longrightarrow \mathcal{D}

equipped with the adjunction unit natural transformation (Def. )

Xη XX X \overset{\eta_X}{\longrightarrow} \bigcirc X

is a modal operator on 𝒟\mathcal{D} (Def. ).

Dually, if

𝒞RAAιAA𝒟 \mathcal{C} \underoverset {\underset{R}{\longleftarrow}} {\overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\bot} \mathcal{D}

is a coreflective subcategory-inclusion (Def. ). Then the composite

ιR:𝒟𝒟 \Box \;\coloneqq\; \iota \circ R \;\colon\; \mathcal{D} \longrightarrow \mathcal{D}

equipped with the adjunction counit natural transformation (Def. )

Xϵ XX \Box X \overset{ \epsilon_X }{\longrightarrow} X

is a comodal operator on 𝒟\mathcal{D} (Def. ).

Conversely:

If an endofunctor :𝒟𝒟\bigcirc \;\colon\; \mathcal{D} \to \mathcal{D} with natural transformation Xη XXX \overset{\eta_X}{\to} \bigcirc X is a modal operator on a category 𝒟\mathcal{D} (Def. ), then the inclusion of its full essential image is a reflective subcategory inclusion (Def. ) with reflector given by the corestriction of \bigcirc to its image:

Im()AAιAA𝒟. Im( \bigcirc ) \underoverset {\underset{ \phantom{AA} \iota \phantom{AA} }{\hookrightarrow}} {\overset{ \bigcirc }{\longleftarrow}} {} \mathcal{D} \,.

Dually, if an endofunctor :𝒟𝒟\Box \;\colon\; \mathcal{D} \to \mathcal{D} with natural transformation Xϵ XX\Box X \overset{\epsilon_X}{\longrightarrow} X is a comodal operator (Def. ), then the inclusion of its full essential image is a coreflective subcategory inclusion (Def. ) with coreflector given by the corestriction of \Box to its image

Im()AAιAA𝒟. Im( \Box ) \underoverset {\underset{ \Box }{\longleftarrow}} {\overset{ \phantom{AA} \iota \phantom{AA} }{\hookrightarrow}} {} \mathcal{D} \,.
Proof

The first two statements are immedialy a special case of the characterization of adjunctions via universal morphisms in Prop. : Using that R=ιR = \iota is here assumed to be fully faithful, the uniqueness of f˜\tilde f in the universal morphism-factorization condition (21)

c η c f R(L(c)) R(f˜) R(d) L(c) !f˜ d \array{ && c \\ & {}^{\mathllap{\eta_c}}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(c)) &&\underset{R (\widetilde f)}{\longrightarrow}&& R(d) \\ \\ L(c) &&\underset{ \exists ! \, \widetilde f}{\longrightarrow}&& d }

implies that also R(f˜)=ι(f˜)R(\widetilde f) = \iota(\widetilde f) is the unique morphism making that triangle commute.

Similarly for the converse: The assumption on a modal operator \bigcirc is just so as to make its unit η\eta be a universal morphism (Def. ) into the inclusion functor ι\iota of its essential image.

Proposition

(modal operator is idempotent)

Let 𝒟\mathcal{D} be a category (Def. ).

For \bigcirc a modal operator on 𝒟\mathcal{D}, with unit η\eta (Def. ), it is idempotent, in that it is naturally isomorphic (Def. ) to the composition with itself:

. \bigcirc \;\simeq\; \bigcirc \bigcirc \,.

In fact, the image under \bigcirc of its unit is such an isomorphism

(Xη XX):X(X) \bigcirc\left( X \overset{\eta_X}{\to} \bigcirc X \right) \;\;\colon\;\; \bigcirc X \overset{\simeq}{\longrightarrow} \bigcirc ( \bigcirc X )

as is its unit on its image

η X:X(X). \eta_{\bigcirc X} \;\;\colon\;\; \bigcirc X \overset{\simeq}{\longrightarrow} \bigcirc ( \bigcirc X ) \,.

Formally dually, for \Box a comodal operator on 𝒟\mathcal{D}, with counit ϵ\epsilon (Def. ), it is idempotent, in that it is naturally isomorphic (Def. ) to the composition with itsef:

. \Box \circ \Box \;\simeq\; \Box \,.

In fact, the image under \Box of its counit is such an isomorphism

(Xϵ XX):(X)X \Box\left( \Box X \overset{\epsilon_X}{\to} X \right) \;\;\colon\;\; \Box (\Box X) \overset{\simeq}{\longrightarrow} \Box X

as is its counit on its image

ϵ X:(X)X. \epsilon_{\Box X} \;\;\colon\;\; \Box ( \Box X ) \overset{\simeq}{\longrightarrow} \Box X \,.
Proof

We discuss the first case, the second is formally dual (Example ).

By Prop. , the modal operator is equivalent to the composite ιL\iota \circ L obtained from the reflective subcategory-inclusion (Def. ) of its essential image of modal objects:

Im()AAιAAAALAA𝒟. Im(\bigcirc) \underoverset {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}} {\overset{\phantom{AA}L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{D} \,.

and its unit is the corresponding adjunction unit (Def. )

Xη Xι(L(X)). X \overset{\eta_X}{\longrightarrow} \iota(L(X)) \,.

Hence it is sufficient to show that the morphisms and L(η X)L( \eta_X ) and η ιY\eta_{\iota Y} are isomorphisms.

Now, the triangle identities (18) for the adjunction LιL \dashv \iota, which hold by Prop. , say that their composition with the adjunction counit is the identity morphism

ϵ L(η X)L(η X)=id L(X)AAandAAι(ϵ Y)η ι(Y)=id ι(Y). \epsilon_{L(\eta_X)} \circ L(\eta_X) \;=\; id_{L(X)} \phantom{AA} \text{and} \phantom{AA} \iota( \epsilon_Y )\circ \eta_{\iota(Y)} \;=\; id_{\iota(Y)} \,.

But by Prop. , the counit ϵ\epsilon is a natural isomorphism, since ι\iota is fully faithful. Hence we may cancel it on both sides of the triangle identities and find that L(η X)L(\eta_X) and η ι(Y)\eta_{\iota(Y)} are indeed isomorphisms.

Definition

(modal objects)

Let 𝒟\mathcal{D} be a category (Def. ).

For \bigcirc a modal operator on 𝒟\mathcal{D} (Def. ), we say:

  1. a \bigcirc-modal object is an object X𝒟X \in \mathcal{D} such that the following conditions hold (which are all equivalent, by Prop. ):

    • it is in the \bigcirc-essential image: XIm()𝒟X \in Im( \bigcirc ) \hookrightarrow \mathcal{D},

    • it is isomorphic to its own \bigcirc-image: XXX \simeq \bigcirc X,

    • specifically its \bigcirc-unit is an isomorphism η X:XX\eta_X \;\colon\; X \overset{\simeq}{\to} \bigcirc X.

  2. a \bigcirc-submodal object is an object X𝒟X \in \mathcal{D}, such that

    • its \bigcirc-unit is a monomorphism (Def. ): η X:XX\eta_X \;\colon\; X \hookrightarrow \bigcirc X.

Dually (Example ):

For \Box a comodal operator on 𝒟\mathcal{D} (Def. ), we say:

  1. a \Box-comodal object is an object X𝒟X \in \mathcal{D} such that the following conditions hold (which are all equivalent, by Prop. ):

    • it is in the \Box-essential image: XIm()𝒟X \in Im( \Box ) \hookrightarrow \mathcal{D},

    • it is isomorphic to its own \Box-image: XX\Box X \simeq X,

    • specifically its \Box-counit is an isomorphism ϵ X:XX\epsilon_X \;\colon\; \Box X \overset{\simeq}{\longrightarrow} X

  2. a \Box-supcomodal object is an object X𝒟X \in \mathcal{D}, such that

    • its \Box-counit is an epimorphism (Def. ): ϵ X:XepiX\epsilon_X \;\colon\; \Box X \overset{epi}{\longrightarrow} X.
Definition

(adjoint modality)

Let

LCR:𝒞ALA ACA ARA𝒟 L \;\dashv\; C \;\dashv\; R \;\colon\; \mathcal{C} \array{ \overset{\phantom{A} L \phantom{A}}{\hookleftarrow} \\ \overset{\phantom{A} C \phantom{A}}{\longrightarrow} \\ \overset{\phantom{A} R \phantom{A}}{\hookleftarrow} } \mathcal{D}

be an adjoint triple (Remark ) such that LL and RR are fully faithful functors (necessarily both, by Prop. ). By Prop. , there are induced modal operators

LCAARC \bigcirc \;\coloneqq\; L \circ C \phantom{AA} \Box \;\coloneqq\; R \circ C

which themselves form am adjoint pair

, \Box \;\dashv\; \bigcirc \,,

hence called an adjoint modality. The adjunction unit and adjunction counit as in (13) may now be read as exhibiting each object XX in the domain of CC as “in between the opposite extremes of its \bigcirc-modal aspect and its \Box-modal aspect”

XAAϵ X AAXAAη X AAX. \Box X \overset{\phantom{AA}\epsilon^\Box_X \phantom{AA}}{\longrightarrow} X \overset{\phantom{AA}\eta^{\bigcirc}_X\phantom{AA}}{\longrightarrow} \bigcirc X \,.

A formally dual situation (Example ) arises when CC is fully faithful.

LCR:𝒞ALA ACA ARA𝒟 L \;\dashv\; C \;\dashv\; R \;\colon\; \mathcal{C} \array{ \overset{\phantom{A} L \phantom{A}}{\longrightarrow} \\ \overset{\phantom{A} C \phantom{A}}{\hookleftarrow} \\ \overset{\phantom{A} R \phantom{A}}{\longrightarrow} } \mathcal{D}

with

(CL)(CR) \left( \bigcirc \;\coloneqq\; C \circ L \right) \;\dashv\; \left( \Box \;\coloneqq\; C \circ R \right)

and canonical natural transformation between opposite extreme aspects given by

(27)XAAϵ X AAXAAη X AAX \Box X \overset{ \phantom{AA} \epsilon^{\Box}_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \eta^{\bigcirc}_X \phantom{AA} }{\longrightarrow} \bigcirc X
Proposition

(fully faithful adjoint triple)

Let LCRL \dashv C \dashv R be an adjoint triple (Remark ). Then the following are equivalent:

  1. LL is a fully faithful functor;

  2. RR is a fully faithful functor,

  3. (LC)(RC)(\Box \;\coloneqq\; L \circ C) \dashv (\bigcirc \;\coloneqq\; R \circ C) is an adjoint modality (Def. ).

For proof see this prop..

In order to analyze (in Prop. below) the comparison morphism of opposite extreme aspects (27) induced by an adjoint modality (Def. ), we need the following technical Lemma:

Lemma

Let

𝒞ALA ACA ARA𝒟 \mathcal{C} \array{ \overset{ \phantom{A} L \phantom{A} }{\longrightarrow} \\ \overset{ \phantom{A} C \phantom{A} }{\hookleftarrow} \\ \overset{ \phantom{A} R \phantom{A} }{ \longrightarrow } } \mathcal{D}

be an adjoint triple with induced adjoint modality (Def. ) to be denoted

(CL)(CR) \left( \bigcirc \;\coloneqq\; C \circ L\right) \;\dashv\; \left( \Box \;\coloneqq\; C \circ R \right)

Denoting the adjunction units/counits (Def. ) as

A\phantom{A} adjunction A\phantom{A}A\phantom{A} unit A\phantom{A}A\phantom{A} counit A\phantom{A}
A\phantom{A} (LC)(L \dashv C) A\phantom{A}A\phantom{A} η \eta^{\bigcirc} A\phantom{A}A\phantom{A} ϵ \epsilon^{\bigcirc} A\phantom{A}
A\phantom{A} (CR)(C \dashv R) A\phantom{A}A\phantom{A} η \eta^\Box A\phantom{A}A\phantom{A} ϵ \epsilon^\Box A\phantom{A}

we have that the following composites of unit/counit components are equal:

(28)(η LX )(Lϵ X )=(Rη X )(ϵ RX )AAAAAALCRX ϵ RX RX Lϵ X Rη X LX η LX RCLX \left( \eta^{\Box}_{L X} \right) \circ \left( L \epsilon^\Box_X \right) \;\;=\;\; \left( R \eta^{\bigcirc}_{X} \right) \circ \left( \epsilon^{\bigcirc}_{R X} \right) \phantom{AAAAAA} \array{ L C R X &\overset{\epsilon^{\bigcirc}_{R X}}{\longrightarrow}& R X \\ {}^{ \mathllap{ L \epsilon^\Box_X } }\big\downarrow && \big\downarrow^{\mathrlap { R \eta^{\bigcirc}_{X} } } \\ L X &\underset{ \eta^\Box_{L X} }{\longrightarrow}& R C L X }

(Johnstone 11, lemma 2.1)

Proof

We claim that the following diagram commutes (Def. ):

RX ϵ RX Rη X LCRX RCLX Lϵ X LCRη X ϵ RCLX η LX LX LCRCLX LX Lη X iso Lϵ CLX LCη LX iso ϵ LX LCLX id LCLX LCLX \array{ && && R X \\ && & {}^{ \epsilon^\bigcirc_{R X} }\nearrow && \searrow^{\mathrlap{ R \eta^{\bigcirc}_X }} \\ && L C R X && && R C L X \\ & {}^{ L \epsilon^\Box_X }\swarrow && \searrow^{ \mathrlap{ L C R \eta^{\bigcirc}_X } } && {}^{\mathllap{ \epsilon^{\bigcirc}_{R C L X} }}\nearrow && \nwarrow^{ \mathrlap{ \eta^{\Box}_{L X} } } \\ L X && && L C R C L X && && L X \\ & {}_{\mathllap{ L \eta^{\bigcirc}_X }}\searrow && {}^{\mathllap{iso}}\swarrow_{\mathrlap{ L \epsilon^{\Box}_{C L X} }} && {}_{\mathllap{ L C \eta^\Box_{L X} }}\nwarrow^{\mathrlap{iso}} && \nearrow_{\mathrlap{ \epsilon^{\bigcirc}_{L X} }} \\ && L C L X && \underset{id_{L C L X}}{\longleftarrow} && L C L X }

This commutes, because:

  1. the left square is the image under LL of naturality (4) for ϵ \epsilon^\Box on η X \eta^{\bigcirc}_X;

  2. the top square is naturality (4) for ϵ \epsilon^{\bigcirc} on Rη X R \eta^{\bigcirc}_X;

  3. the right square is naturality (4) for ϵ \epsilon^{\bigcirc} on η LX \eta^{\Box}_{L X};

  4. the bottom commuting triangle is the image under LL of the triangle identity (18) for (CR)(C \dashv R) on LXL X.

Moreover, notice that

  1. the total bottom composite is the identity morphism id LXid_{L X}, due to the triangle identity (18) for (CR)(C \dashv R);

  2. also the other two morphisms in the bottom triangle are isomorphisms, as shown, due to the idempoency of the (CR)(C-R)-adjunction (Prop. .)

Therefore the total composite from LCRXR/CLXL C R X \to R/ C L X along the bottom part of the diagram equals the left hand side of (28), while the composite along the top part of the diagram clearly equals the right hand side of (28).

Proposition

(comparison transformation between opposite extremes of adjoint modality)

Consider an adjoint triple of the form

LCR:𝒞AALAA AACAA AAARAAA L \dashv C \dashv R \;\;\colon\;\; \mathcal{C} \array{ \overset{\phantom{AA} L \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} C \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} R \phantom{AAA} }{\longrightarrow} } \mathcal{B}

with induced adjoint modality (Def. ) to be denoted

(CL)(CR) \left( \bigcirc \;\coloneqq\; C \circ L \right) \;\dashv\; \left( \Box \;\coloneqq\; C \circ R \right)

Denoting the adjunction units/counits (Def. ) as

A\phantom{A} adjunction A\phantom{A}A\phantom{A} unit A\phantom{A}A\phantom{A} counit A\phantom{A}
A\phantom{A} (LC)(L \dashv C) A\phantom{A}A\phantom{A} η \eta^{\bigcirc} A\phantom{A}A\phantom{A} ϵ \epsilon^{\bigcirc} A\phantom{A}
A\phantom{A} (CE)(C \dashv E) A\phantom{A}A\phantom{A} η \eta^\Box A\phantom{A}A\phantom{A} ϵ \epsilon^\Box A\phantom{A}

Then for all X𝒞X \in \mathcal{C} the following two natural transformations, constructed from the adjunction units/counits (Def. ) and their inverse morphisms (using idempotency, Prop. ), are equal:

(29)comp (Lϵ X )(η RX ) 1=(η LX ) 1(Γη X )AAAAAAARX Γη X RCLX (η RX ) 1 comp (η LX ) 1 LCRX Lϵ X LX comp_{\mathcal{B}} \;\;\coloneqq\;\; \left( L \epsilon^\Box_X \right) \circ \left( \eta^{\bigcirc}_{R X} \right)^{-1} \;\;=\;\; \left( \eta^\Box_{L X} \right)^{-1} \circ \left( \Gamma \eta^{\bigcirc}_X \right) \phantom{AAAAAAA} \array{ R X & \overset{ \Gamma \eta^{\bigcirc}_X }{\longrightarrow} & R C L X \\ {}^{ \mathllap{ \left( \eta^{\bigcirc}_{R X} \right)^{-1} } }\big\downarrow & \searrow^{ { comp_{\mathcal{B}} } } & \big\downarrow^{ \mathrlap{ \left( \eta^\Box_{L X} \right)^{-1} } } \\ L C R X &\underset{ L \epsilon^\Box_X }{\longrightarrow}& L X }

Moreover, the image of these morphisms under CC equals the following composite:

(30)comp 𝒞:XAϵ X AXAη X AX, comp_{\mathcal{C}} \;\colon\; \Box X \overset{ \phantom{A} \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} \bigcirc X \,,

hence

(31)comp 𝒞=C(comp ). comp_{\mathcal{C}} \;=\; C(comp_{\mathcal{B}}) \,.
Proof

The first statement follows directly from Lemma .

For the second statement, notice that the (CR)(C \dashv R)-adjunct (Prop. ) of

comp 𝒞:CRXAϵ X AXAη X ACLX comp_{\mathcal{C}} \;\colon\; C R X \overset{ \phantom{A} \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} X \overset{ \phantom{A} \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} C L X

is

(32)comp 𝒞˜=ΓXisoAη RX ARCRXisoAΓϵ X ARX=id RXARη X ARCLX, \widetilde{ comp_{\mathcal{C}} } \;\;=\;\; \underset{ = id_{R X} }{ \underbrace{ \Gamma X \underoverset{iso}{ \phantom{A} \eta^{\Box}_{R X} \phantom{A} }{ \longrightarrow } R C R X \underoverset{iso}{ \phantom{A} \Gamma \epsilon^{\Box}_X \phantom{A} }{\longrightarrow} R X }} \overset{ \phantom{A} R \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow} R C L X \,,

where under the braces we uses the triangle identity (Prop. ).

(As a side remark, for later usage, we observe that the morphisms on the left in (32) are isomorphisms, as shown, by idempotency of the adjunctions.)

From this we obtain the following commuting diagram:

CRX ACRη X A CRCLX isoAC(η LX ) 1A CLX comp 𝒞 ϵ CLX id LX CLX \array{ C R X &\overset{ \phantom{A} C R \eta^{\bigcirc}_X \phantom{A} }{\longrightarrow}& C R C L X &\underoverset{iso}{ \phantom{A} C \left(\eta^{ \Box }_{L X}\right)^{-1} \phantom{A} }{ \longrightarrow }& C L X \\ &{}_{\mathllap{ comp_{\mathcal{C}} }}\searrow& {}^{ \mathllap{ \epsilon^{\Box}_{C L X} } } \big\downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{id_{L X}}} \\ && C L X }

Here:

  1. on the left we identified comp 𝒞˜˜=comp 𝒞\widetilde {\widetilde {comp_{\mathcal{C}}}} \;=\; comp_{\mathcal{C}} by applying the formula (Prop. ) for (CR)(C \dashv R)-adjuncts to comp 𝒞˜=Rη X \widetilde {comp_{\mathcal{C}}} = R \eta^{\bigcirc}_X (32);

  2. on the right we used the triangle identity (Prop. ) for (CR)(C \dashv R).

This proves the second statement.

Definition

(preorder on modalities)

Let 1\bigcirc_1 and 2\bigcirc_2 be two modal operators on a category 𝒞\mathcal{C}. By Prop. these are equivalently characterized by their reflective full subcategories 𝒞 1,𝒞 2𝒞\mathcal{C}_{\bigcirc_1}, \mathcal{C}_{\bigcirc}_2 \hookrightarrow \mathcal{C} of modal objects.

There is an evident preorder on full subcategories of 𝒞\mathcal{C}, given by full inclusions of full subcategories into each other. We write 𝒞 1𝒞 2\mathcal{C}_{\bigcirc_1} \subset \mathcal{C}_{\bigcirc_2} if the full subcategory on the left is contained, as a full subcategory of 𝒞\mathcal{C}, in that on the right. Via prop. there is the induced preorder on modal operators, and we write

1< 2AAiffAA𝒞 1𝒞 2. \bigcirc_1 \;\lt\; \bigcirc_2 \phantom{AA} iff \phantom{AA} \mathcal{C}_{\bigcirc_1} \;\subset\; \mathcal{C}_{\bigcirc_2} \,.

There is an analogous preorder on comodal operators (Def. ).

If we have two adjoint modalities (Def. ) of the same type (both modal left adjoint or both comodal left adjoint) such that both the modalities and the comodalities are compatibly ordered in this way, we denote this situation as follows:

2 2 1 1AAAAorAAAA 2 2 1 1 \array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee && \vee \\ \bigcirc_1 &\dashv& \Box_1 } \phantom{AAAA} \text{or} \phantom{AAAA} \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }

etc.

Example

(bottom and top adjoint modality)

Let 𝒞\mathcal{C} be a category with both an initial object \emptyset and a terminal object *\ast (Def. ). Then, by Example there is an adjoint triple between 𝒞\mathcal{C} and the terminal category *\ast (Example ) of the form

𝒞Aconst A AAAA Aconst *A*. \mathcal{C} \array{ \overset{ \phantom{A} const_\emptyset \phantom{A} }{\hookleftarrow} \\ \overset{\phantom{AAAA}}{\longrightarrow} \\ \overset{ \phantom{A} const_\ast \phantom{A} }{\hookleftarrow} } \ast \,.

The induced adjoint modality (Def. ) is

const const *:𝒞𝒞. const_{\emptyset} \;\dashv\; const_\ast \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.

By slight abuse of notation, we will also write this as

(33)*:𝒞𝒞. \emptyset \;\dashv\; \ast \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.

On the other extreme, for 𝒞\mathcal{C} any category whatsoever, the identity functor on it is adjoint functor to itself, and constitutes an adjoint modality (Def. )

(34)id 𝒞id 𝒞:𝒞𝒞. id_{\mathcal{C}} \;\dashv\; id_{\mathcal{C}} \;\;\colon\;\; \mathcal{C} \to \mathcal{C} \,.

Here

  1. (33) is the bottom (or ground)

  2. (34) is the top

in the preorder on adjoint modalities according to Def. , in that for every adjoint modality of the form \bigcirc \dashv \Box we have the following:

id id * \array{ id &\dashv& id \\ \vee && \vee \\ \Box &\dashv& \bigcirc \\ \vee && \vee \\ \emptyset &\dashv& \ast }
Definition

(Aufhebung)

On some category 𝒞\mathcal{C}, consider an inclusion of adjoint modalities, according to Def. :

2 2 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee && \vee \\ \Box_1 &\dashv& \bigcirc_1 }

We say:

  1. This provides right Aufhebung of the opposition exhibited by 1 1\Box_1 \dashv \bigcirc_1 if there is also the diagonal inclusion

    1< 2AAAequivalentlyAAA𝒞 1𝒞 2 \Box_1 \lt \bigcirc_2 \phantom{AAA} equivalently \phantom{AAA} \mathcal{C}_{\Box_1} \subset \mathcal{C}_{\bigcirc_2}

    We indicate this situation by

    2 2 / 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &/& \vee \\ \Box_1 &\dashv& \bigcirc_1 }
  2. This provides left Aufhebung of the opposition exhibited by 1 1\Box_1 \dashv \bigcirc_1 if there is also the diagonal inclusion

    1< 2AAAequivalentlyAAA𝒞 1𝒞 2 \bigcirc_1 \lt \Box_2 \phantom{AAA} equivalently \phantom{AAA} \mathcal{C}_{\bigcirc_1} \subset \mathcal{C}_{\Box_2}

    We indicate this situation by

    2 2 \ 1 1 \array{ \Box_2 &\dashv& \bigcirc_2 \\ \vee &\backslash& \vee \\ \Box_1 &\dashv& \bigcirc_1 }
Remark

For a progression of adjoint modalities of the form

2 2 1 1 \array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee && \vee \\ \bigcirc_1 &\dashv& \Box_1 }

the analog of Aufhebung (Def. ) is automatic, since, by Prop. , in this situation the full subcategories modal objects at each stage coincide already.

For emphasis we may denote this situation by

2 2 | 1 1. \array{ \bigcirc_2 &\dashv& \Box_2 \\ \vee &\vert& \vee \\ \bigcirc_1 &\dashv& \Box_1 } \,.
Example

(top adjoint modality provides Aufhebung of all oppositions)

For 𝒞\mathcal{C} any category, the top adjoint modality ididid \dashv id (Def. ) provides Aufhebung (Def. ) of every other adjoint modality.

But already Aufhebung of the bottom adjoint modality is a non-trivial and interesting condition. We consider this below in Prop. .

\,

We now re-consider the concept of reflective subcategories from the point of view of localization of categories:

Definition

(category with weak equivalences)

A category with weak equivalences is

  1. a category 𝒞\mathcal{C} (Def. )

  2. a subcategory W𝒞W \subset \mathcal{C} (i.e. sub-class of objects and morphisms that inherits the structure of a category)

such that the morphisms in WW

  1. include all the isomorphisms of 𝒞\mathcal{C},

  2. satisfy two-out-of-three:

    If for gg, ff any two composable morphisms in 𝒞\mathcal{C}, two out of the set {g,f,gf}\{g,\, f,\, g \circ f \} are in WW, then so is the third.

    f g gf \array{ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ && \underset{ g \circ f }{\longrightarrow} }
Definition

(localization of a category)

Let W𝒞W \subset \mathcal{C} be a category with weak equivalences (Def. ). Then the localization of 𝒞\mathcal{C} at WW is, if it exsists

  1. a category 𝒞[W 1]\mathcal{C}[W^{-1}],

  2. a functor γ:𝒞𝒞[W 1]\gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}] (Def. )

such that

  1. γ\gamma sends all morphisms in W𝒞W \subset \mathcal{C} to isomorphisms (Def. ),

  2. γ\gamma is universal with this property: If F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} is any functor with this property, then it factors through γ\gamma, up to natural isomorphism (Def. ):

    FDFγAAAAAAA𝒞 F 𝒟 γ ρ DF 𝒞[W 1] F \;\simeq\; D F \circ \gamma \phantom{AAAAAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}[W^{-1}] }

    and any two such factorizations DFD F and D FD^' F are related by a unique natural isomorphism κ\kappa compatible with ρ\rho and ρ \rho^':

(35)𝒞 F 𝒟 γ ρ DF id 𝒞[W 1] κ 𝒟 id D F 𝒞[W 1]AAAA=AAAA𝒞 F 𝒟 γ ρ D F 𝒞[W 1] \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D F}} && \searrow^{\mathrlap{id}} \\ && \mathcal{C}[W^{-1}] && {}_{\simeq}\seArrow^{\kappa} && \mathcal{D} \\ && & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{D^' F}} \\ && && \mathcal{C}[W^{-1}] } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{\gamma}}\searrow &{}^{\rho^'}\Downarrow_{\simeq}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}[W^{-1}] }

Such a localization is called a reflective localization if the localization functor has a fully faithful right adjoint, exhibiting it as the reflection functor of a reflective subcategory-inclusion (Def. )

𝒞[W 1]AAAAAAγAA𝒞. \mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{ \phantom{AA} \gamma \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C} \,.
Proposition

(reflective subcategories are localizations)

Every reflective subcategory-inclusion (Def. )

𝒞 LAAιAAAALAA𝒞 \mathcal{C}_{L} \underoverset {\underset{\phantom{AA}\iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}

is the reflective localization (Def. ) at the class WL 1(Isos)W \coloneqq L^{-1}(Isos) of morphisms that are sent to isomorphisms by the reflector LL.

Proof

Let F:𝒞𝒟F \;\colon\; \mathcal{C} \to \mathcal{D} be a functor which inverts morphisms that are inverted by LL.

First we need to show that it factors through LL, up to natural isomorphism. But consider the following whiskering of the adjunction unit η\eta (Def. ) with FF:

𝒞 F 𝒟 L DF 𝒞 LAAAA𝒞 id 𝒞 F 𝒟 L η ι 𝒞 L \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow& \nearrow_{\mathrlap{D F}} \\ && \mathcal{C}_L } \phantom{AA} \coloneqq \phantom{AA} \array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} & \overset{F}{\longrightarrow}& \mathcal{D} \\ & {}_{\mathllap{L}}\searrow &\Downarrow^{\eta}& \nearrow_{\mathrlap{\iota}} \\ && \mathcal{C}_L }

By idempotency (Prop. ), the components of the adjunction unit η\eta are inverted by LL, and hence by assumption they are also inverted by FF, so that on the right the natural transformation F(η)F(\eta) is indeed a natural isomorphism.

It remains to show that this factorization is unique up to unique natural isomorphism. So consider any other factorization D FD^' F via a natural isomorphism ρ\rho. Pasting this now with the adjunction counit

𝒞 F 𝒟 ι ϵ L ρ D F 𝒞 L id 𝒞 L \array{ && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L }

exhibits a natural isomorphism ϵρ\epsilon \cdot \rho between DFD FD F \simeq D^' F. Moreover, this is compatible with F(η)F(\eta) according to (35), due to the triangle identity (Prop. ):

𝒞 id 𝒞 F 𝒟 L η ι ϵ L ρ D F 𝒞 L id 𝒞 LAAAA=AAAA𝒞 F 𝒟 ρ 𝒞 L \array{ \mathcal{C} && \overset{id}{\longrightarrow} && \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & {}_{\mathllap{L}}\searrow & {}^{\mathllap{\eta}}\Downarrow & {}^{\mathllap{\iota}}\nearrow & {}^{\epsilon}\Downarrow & {}_{\mathllap{L}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{D^' F}} \\ && \mathcal{C}_L && \underset{ id }{\longrightarrow} && \mathcal{C}_L } \phantom{AAAA} = \phantom{AAAA} \array{ \mathcal{C} && \overset{F}{\longrightarrow} && \mathcal{D} \\ & \searrow &\Downarrow^\rho& \swarrow \\ && \mathcal{C}_L }

Finally, since LL is essentially surjective functor, by idempotency (Prop. ), it is clear that this is the unique such natural isomorphism.

Definition

(local object)

Let 𝒞\mathcal{C} be a category (Def. ) and let SMor 𝒞S \subset Mor_{\mathcal{C}} be a set of morphisms. Then an object X𝒞X \in \mathcal{C} is called an SS-local object if for all AsBSA \overset{s}{\to} B \; \in S the hom-functor (Def. ) from ss into XX yields a bijection

Hom 𝒞(s,X):Hom 𝒞(B,X)AAAAHom 𝒞(A,X), Hom_{\mathcal{C}}(s,X) \;\colon\; Hom_{\mathcal{C}}(B,X) \overset{ \phantom{AA} \simeq \phantom{AA} }{\longrightarrow} Hom_{\mathcal{C}}(A,X) \,,

hence if every morphism AfXA \overset{f}{\longrightarrow} X extends uniquely along ww to BB:

A AfA X w ! B \array{ A &\overset{\phantom{A}f\phantom{A}}{\longrightarrow}& X \\ {}^{\mathllap{w}}\big\downarrow & \nearrow_{\mathrlap{ \exists! }} \\ B }

We write

(36)𝒞 SAAιAA𝒞 \mathcal{C}_S \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow} \mathcal{C}

for the full subcategory (Example ) of SS-local objects.

Definition

(reflection onto full subcategory of local objects)

Let 𝒞\mathcal{C} be a category and set SMor 𝒞S \subset Mor_{\mathcal{C}} be a sub-class of its morphisms. Then the reflection onto local SS-objects (often just called “localization at the collection SS” is, if it exists, a left adjoint (Def. ) LL to the full subcategory-inclusion of the SS-local objects (36):

𝒞 SιAALAA𝒞. \mathcal{C}_S \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C} \,.

A class of examples is the following, which comes to its full nature (only) after passage to homotopy theory (Example below):

Definition

(homotopy localization of 1-categories)

Let 𝒞\mathcal{C} be a category, let 𝔸𝒞\mathbb{A} \in \mathcal{C} be an object, and consider the class of morphisms given by projection out of the Cartesian product with 𝔸\mathbb{A}, of all objects X𝒞X \in \mathcal{C}:

X×𝔸p 1X. X \times \mathbb{A} \overset{p_1}{\longrightarrow} X \,.

If the corresponding reflection onto the full subcategory of local objects (Def. ) exists, we say this is homotopy localization at that object , and denote the modal operator corresponding to this (via Prop. ) by

𝔸:𝒞𝒞. \bigcirc\!\!\!\!\!\!\!\!\mathbb{A} \;\colon\; \mathcal{C} \longrightarrow \mathcal{C} \,.
Proposition

(reflective localization reflects onto full subcategory of local objects)

Let W𝒞W \subset \mathcal{C} be a category with weak equivalences (Def. ). If its reflective localization (Def. ) exists

𝒞[W 1]AAιAAAALAA𝒞 \mathcal{C}[W^{-1}] \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{ \phantom{AA} L \phantom{AA} }{\longleftarrow}} {\bot} \mathcal{C}

then 𝒞[W 1]ι𝒞\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C} is equivalently the inclusion of the full subcategory (Example ) on the WW-local objects (Def. ), and hence LL is equivalently reflection onto the WW-local objects, according to Def. .

Proof

We need to show that

  1. every X𝒞[W 1]ι𝒞X \in \mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C} is WW-local,

  2. every Y𝒞Y \in \mathcal{C} is WW-local precisely if it is isomorphic to an object in 𝒞[W 1]ι𝒞\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}.

The first statement follows directly with the adjunction isomorphism (10):

Hom 𝒞(w,ι(X))Hom 𝒞[W 1](L(w),X) Hom_{\mathcal{C}}(w, \iota(X)) \simeq Hom_{\mathcal{C}[W^{-1}]}(L(w), X)

and the fact that the hom-functor takes isomorphisms to bijections (Example ).

For the second statement, consider the case that YY is WW-local. Observe that then YY is also local with respect to the class

W satL 1(Isos) W_{sat} \;\coloneqq\; L^{-1}(Isos)

of all morphisms that are inverted by LL (the “saturated class of morphisms”): For consider the hom-functor 𝒞Hom 𝒞(,Y)Set op\mathcal{C} \overset{Hom_{\mathcal{C}}(-,Y)}{\longrightarrow} Set^{op} to the opposite of the category of sets. By assumption on YY this takes elements in WW to isomorphisms. Hence, by the defining universal property of the localization-functor LL, it factors through LL, up to natural isomorphism.

Since, by idempotency (Prop. ), the adjunction unit η Y\eta_Y is in W satW_{sat}, this implies that we have a bijection of the form

Hom 𝒞(η Y,Y):Hom 𝒞(ιL(Y),Y)Hom 𝒞(Y,Y). Hom_{\mathcal{C}}( \eta_Y, Y ) \;\colon\; Hom_{\mathcal{C}}( \iota L(Y), Y ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}(Y, Y) \,.

In particular the identity morphism id Yid_Y has a preimage η Y 1\eta_Y^{-1} under this function, hence a left inverse to η\eta:

η Y 1η Y=id Y. \eta_Y^{-1} \circ \eta_Y \;=\; id_Y \,.

But by 2-out-of-3 this implies that η Y 1W sat\eta_Y^{-1} \in W_{sat}. Since the first item above shows that ιL(Y)\iota L(Y) is W satW_{sat}-local, this allows to apply this same kind of argument again,

Hom 𝒞(η Y 1,ιL(Y)):Hom 𝒞(Y,ιL(Y))Hom 𝒞(ιL(Y),ιL(Y)), Hom_{\mathcal{C}}( \eta^{-1}_Y, \iota L(Y) ) \;\colon\; Hom_{\mathcal{C}}( Y, \iota L(Y) ) \overset{\simeq}{\longrightarrow} Hom_{\mathcal{C}}( \iota L(Y) , \iota L(Y)) \,,

to deduce that also η Y 1\eta_Y^{-1} has a left inverse (η Y 1) 1η Y 1(\eta_Y^{-1})^{-1} \circ \eta_Y^{-1}. But since a left inverse that itself has a left inverse is in fact an inverse morphisms (this Lemma), this means that η Y 1\eta^{-1}_Y is an inverse morphism to η Y\eta_Y, hence that η Y:YιL(Y)\eta_Y \;\colon\; Y \to \iota L (Y) is an isomorphism and hence that YY is isomorphic to an object in 𝒞[W 1]ι𝒞\mathcal{C}[W^{-1}] \overset{\iota}{\hookrightarrow} \mathcal{C}.

Conversely, if there is an isomorphism from YY to a morphism in the image of ι\iota hence, by the first item, to a WW-local object, it follows immediatly that also YY is WW-local, since the hom-functor takes isomorphisms to bijections and since bijections satisfy 2-out-of-3.

Proposition

(reflection onto local objects is localization with respect to left adjoints)

Let 𝒞\mathcal{C} be a category (Def. ) and let SMor 𝒞S \subset Mor_{\mathcal{C}} be a class of morphisms in 𝒞\mathcal{C}. Then the reflection onto the SS-local objects (Def. ) satisfies, if it exists, the universal property of a localization of categories (Def. ) with respect to left adjoint functors inverting SS.

Proof

Write

𝒞 SAAιAAAALAA𝒞 \mathcal{C}_S \underoverset {\underset{ \phantom{AA}\iota\phantom{AA} }{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{C}

for the reflective subcategory-inclusion of the SS-local objects.

Say that a morphism ff in 𝒞\mathcal{C} is an SS-local morphism if for every SS-local object A𝒞A \in \mathcal{C} the hom-functor (Example ) from ff to AA yields a bijection Hom 𝒞(f,A)Hom_{\mathcal{C}}(f,A). Notice that, by the Yoneda embedding for 𝒞 S\mathcal{C}_S (Prop. ), the SS-local morphisms are precisely the morphisms that are taken to isomorphisms by the reflector LL (via Example ).

Now let

(FG):𝒞GAAFAA𝒟 (F \dashv G) \;\colon\; \mathcal{C} \underoverset {\underset{G}{\longleftarrow}} {\overset{ \phantom{AA} F \phantom{AA} }{\longrightarrow}} {\bot} \mathcal{D}

be a pair of adjoint functors, such that the left adjoint FF inverts the morphisms in SS. By the adjunction hom-isomorphism (10) it follows that GG takes values in SS-local objects. This in turn implies, now via the Yoneda embedding for 𝒟\mathcal{D}, that FF inverts all SS-local morphisms, and hence all morphisms that are inverted by LL.

Thus the essentially unique factorization of FF through LL now follows by Prop. .

\,

Basic notions of Categorical algebra

We have seen that the existence of Cartesian products in a category 𝒞\mathcal{C} equips is with a functor of the form

𝒞×𝒞()×()𝒞 \mathcal{C} \times \mathcal{C} \overset{ (-) \times (-) }{\longrightarrow} \mathcal{C}

which is directly analogous to the operation of multiplication in an associative algebra or even just in a semigroup (or monoid), just “categorified” (Example below). This is made precise by the concept of a monoidal category (Def. below).

This relation between category theory and algebra leads to the fields of categorical algebra and of universal algebra.

Here we are mainly interested in monoidal categories as a foundations for enriched category theory, to which we turn below.

\,

Monoidal categories

Definition

(monoidal category)

An_monoidal category_ is a category 𝒞\mathcal{C} (Def. ) equipped with

  1. a functor (Def. )

    :𝒞×𝒞𝒞 \otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C}

    out of the product category of 𝒞\mathcal{C} with itself (Example ), called the tensor product,

  2. an object

    1Obj 𝒞 1 \in Obj_{\mathcal{C}}

    called the unit object or tensor unit,

  3. a natural isomorphism (Def. )

    a:(()())()()(()()) a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-))

    called the associator,

  4. a natural isomorphism

    :(1())() \ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-)

    called the left unitor, and a natural isomorphism

    r:()1() r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-)

    called the right unitor,

such that the following two kinds of diagrams commute, for all objects involved:

  1. triangle identity:

    (x1)y a x,1,y x(1y) ρ x1 y 1 xλ y xy \array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && }
  2. the pentagon identity:

    (wx)(yz) α wx,y,z α w,x,yz ((wx)y)z (w(x(yz))) α w,x,yid z id wα x,y,z (w(xy))z α w,xy,z w((xy)z) \array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) }
Example

(cartesian monoidal category)

Let 𝒞\mathcal{C} be a category in which all finite products exist. Then 𝒞\mathcal{C} becomes a monoidal category (Def. ) by

  1. taking the tensor product to be the Cartesian product

    XYX×Y X \otimes Y \;\coloneqq\; X \times Y
  2. taking the unit object to be the terminal object (Def. )

    I* I \;\coloneqq\; \ast

Monoidal categories of this form are called cartesian monoidal categories.

Lemma

(Kelly 64)

Let (𝒞,,1)(\mathcal{C}, \otimes, 1) be a monoidal category, def. . Then the left and right unitors \ell and rr satisfy the following conditions:

  1. 1=r 1:111\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1;

  2. for all objects x,y𝒞x,y \in \mathcal{C} the following diagrams commutes:

    (1x)y α 1,x,y xid y 1(xy) xy xy; \array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

    and

    x(y1) α 1,x,y 1 id xr y (xy)1 r xy xy; \array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,;

For proof see at monoidal category this lemma and this lemma.

Remark

Just as for an associative algebra it is sufficient to demand 1a=a1 a = a and a1=aa 1 = a and (ab)c=a(bc)(a b) c = a (b c) in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a coherence theorem for monoidal categories which states that all ways of freely composing the unitors and associators in a monoidal category (def. ) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an associative algebra altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.

(Here the qualifier “freely” means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a strong monoidal functor from a free monoidal category. For example if in a particular monoidal category it so happens that the object X(YZ)X \otimes (Y \otimes Z) is actually equal to (XY)Z(X \otimes Y)\otimes Z, then the various ways of going from one expression to another using only associators and this equality no longer need to coincide.)

Definition

(braided monoidal category)

A braided monoidal category, is a monoidal category 𝒞\mathcal{C} (def. ) equipped with a natural isomorphism (Def. )

(37)τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

called the braiding, such that the following two kinds of diagrams commute for all objects involved (“hexagon identities”):

(xy)z a x,y,z x(yz) τ x,yz (yz)x τ x,yId a y,z,x (yx)z a y,x,z y(xz) Idτ x,z y(zx) \array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) }

and

x(yz) a x,y,z 1 (xy)z τ xy,z z(xy) Idτ y,z a z,x,y 1 x(zy) a x,z,y 1 (xz)y τ x,zId (zx)y, \array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,,

where a x,y,z:(xy)zx(yz)a_{x,y,z} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z) denotes the components of the associator of 𝒞 \mathcal{C}^\otimes.

Definition

A symmetric monoidal category is a braided monoidal category (def. ) for which the braiding

τ x,y:xyyx \tau_{x,y} \colon x \otimes y \to y \otimes x

satisfies the condition:

τ y,xτ x,y=1 xy \tau_{y,x} \circ \tau_{x,y} = 1_{x \otimes y}

for all objects x,yx, y

Remark

In analogy to the coherence theorem for monoidal categories (remark ) there is a coherence theorem for symmetric monoidal categories (def. ), saying that every diagram built freely (see remark ) from associators, unitors and braidings such that both sides of the diagram correspond to the same permutation of objects, coincide.

Definition

(symmetric closed monoidal category)

Given a symmetric monoidal category 𝒞\mathcal{C} with tensor product \otimes (def. ) it is called a closed monoidal category if for each Y𝒞Y \in \mathcal{C} the functor Y()()YY \otimes(-)\simeq (-)\otimes Y has a right adjoint, denoted hom(Y,)hom(Y,-)

(38)𝒞[Y,]()Y𝒞, \mathcal{C} \underoverset {\underset{ [Y,-]}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,,

hence if there are natural bijections

Hom 𝒞(XY,Z)Hom 𝒞C(X,[Y,Z]) Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, [Y,Z])

for all objects X,Z𝒞X,Z \in \mathcal{C}.

Since for the case that X=1X = 1 is the tensor unit of 𝒞\mathcal{C} this means that

Hom 𝒞(1,[Y,Z])Hom 𝒞(Y,Z), Hom_{\mathcal{C}}(1, [Y,Z]) \simeq Hom_{\mathcal{C}}(Y,Z) \,,

the object [Y,Z]𝒞[Y,Z] \in \mathcal{C} is an enhancement of the ordinary hom-set Hom 𝒞(Y,Z)Hom_{\mathcal{C}}(Y,Z) to an object in 𝒞\mathcal{C}. Accordingly, it is also called the internal hom between YY and ZZ.

The adjunction counit (Def. ) in this case is called the evaluation morphism

(39)X[X,Y]evY X \otimes [X,Y] \overset{ev}{\longrightarrow} Y
Example

(Set is a cartesian closed category)

The category Set of all sets (Example ) equipped with its cartesian monoidal category-structure (Example ) is a closed monoidal category (Def. ), hence a cartesian closed category. The Cartesian product is the original Cartesian product of sets, and the internal hom is the function set [X,Y][X,Y] of functions from XX to YY

Example

(tensor product of abelian groups is closed monoidal category symmetric monoidal category-structure)

The category Ab of abelian groups (as in Example ) becomes a symmetric monoidal category (Def. ) with tensor product the actual tensor product of abelian groups \otimes_{\mathbb{Z}} and with tensor unit the additive group \mathbb{Z} of integers. Again the associator, unitor and braiding isomorphism are the evident ones coming from the underlying sets.

This is a closed monoidal category with internal hom hom(A,B)hom(A,B) being the set of homomorphisms Hom Ab(A,B)Hom_{Ab}(A,B) equipped with the pointwise group structure for ϕ 1,ϕ 2Hom Ab(A,B)\phi_1, \phi_2 \in Hom_{Ab}(A,B) then (ϕ 1+ϕ 2)(a)ϕ 1(a)+ϕ 2(b)B(\phi_1 + \phi_2)(a) \coloneqq \phi_1(a) + \phi_2(b) \; \in B.

This is the archetypical case that motivates the notation “\otimes” for the pairing operation in a monoidal category.

Example

(Cat and Grpd are cartesian closed categories)

The category Cat (Example ) of all small categories (Example ) is a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

Inside this, the full subcategory (Example ) Grpd (Example ) of all small groupoids (Example ) is itself a cartesian monoidal category-structure (Example ) with Cartesian product given by forming product categories (Example ).

In both cases this yields a closed monoidal category (Def. ), hence a cartesian closed category: the internal hom is given by the functor category construction (Example ).

Example

(categories of presheaves are cartesian closed)

Let 𝒞\mathcal{C} be a category and write [𝒞 op,Set][\mathcal{C}^{op}, Set] for its category of presheaves (Example ).

This is

  1. a cartesian monoidal category (Example ), whose Cartesian product is given objectwise in 𝒞\mathcal{C} by the Cartesian product in Set:

    for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set], their Cartesian product X×Y\mathbf{X} \times \mathbf{Y} exists and is given by

    X×Y:Ac 1 X(c 1)×Y(c 1) f X(f)×Y(f) c 2 X(c 2)×Y(c 2) \mathbf{X} \times \mathbf{Y} \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& \mathbf{X}(c_1) \times \mathbf{Y}(c_1) \\ {}^{\mathllap{f}}\big\downarrow && \big\uparrow^{ \mathrlap{ \mathbf{X}(f) \times \mathbf{Y}(f) } } \\ c_2 &\mapsto& \mathbf{X}(c_2) \times \mathbf{Y}(c_2) }
  2. a cartesian closed category (Def. ), whose internal hom is given for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] by

    [X,Y]:Ac 1 Hom [𝒞 op,Set](y(c 1)×X,y) f Hom [𝒞 op,Set](y(f)×X,y) c 2 Hom [𝒞 op,Set](y(c 2)×X,y) [\mathbf{X}, \mathbf{Y}] \;\;\colon\;\;\phantom{A} \array{ c_1 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_1) \times \mathbf{X}, \mathbf{y} ) \\ {}^{ \mathllap{ f } }\big\downarrow && \big\uparrow^{ \mathrlap{ Hom_{[\mathcal{C}^{op}, Set]}( y(f) \times \mathbf{X}, \mathbf{y} ) } } \\ c_2 &\mapsto& Hom_{[\mathcal{C}^{op}, Set]}( y(c_2) \times \mathbf{X}, \mathbf{y} ) }

    Here y:𝒞[𝒞 op,Set]y \;\colon\; \mathcal{C} \to [\mathcal{C}^{op}, Set] denotes the Yoneda embedding and Hom [𝒞 op,Set](,)Hom_{[\mathcal{C}^{op}, Set]}(-,-) is the hom-functor on the category of presheaves.

Proof

The first statement is a special case of the general fact that limits of presheaves are computed objectwise (Example ).

For the second statement, first assume that [X,Y][\mathbf{X}, \mathbf{Y}] does exist. Then by the adjunction hom-isomorphism (10) we have for any other presheaf Z\mathbf{Z} a natural isomorphism of the form

(40)Hom [𝒞 op,Set](Z,[X,Y])Hom [𝒞 op,Set](Z×X,Y). Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z}, [\mathbf{X},\mathbf{Y}]) \;\simeq\; Hom_{[\mathcal{C}^{op}, Set]}(\mathbf{Z} \times \mathbf{X}, \mathbf{Y}) \,.

This holds in particular for Z=y(c)\mathbf{Z} = y(c) a representable presheaf (Example ) and so the Yoneda lemma (Prop. ) implies that if it exists, then [X,Y][\mathbf{X}, \mathbf{Y}] must have the claimed form:

[X,Y](c) Hom [𝒞 op,Set](y(c),[X,Y]) Hom [𝒞 op,Set](y(c)×X,Y). \begin{aligned} [\mathbf{X}, \mathbf{Y}](c) & \simeq Hom_{[\mathcal{C}^{op}, Set]}( y(c), [\mathbf{X}, \mathbf{Y}] ) \\ & \simeq Hom_{ [\mathcal{C}^{op}, Set] }( y(c) \times \mathbf{X}, \mathbf{Y} ) \,. \end{aligned}

Hence it remains to show that this formula does make (40) hold generally.

For this we use the equivalent characterization of adjoint functors from Prop. , in terms of the adjunction counit providing a system of universal arrows (Def. ).

Define a would-be adjunction counit, hence a would-be evaluation morphism (39), by

X×[X,Y] ev Y X(c)×Hom [𝒞 op,Set](y(c)×X,Y) ev c Y(c) (x,ϕ) ϕ c(id c,x) \array{ \mathbf{X} \times [\mathbf{X} , \mathbf{Y}] &\overset{ev}{\longrightarrow}& \mathbf{Y} \\ \mathbf{X}(c) \times Hom_{[\mathcal{C}^{op}, Set]}(y(c) \times \mathbf{X}, \mathbf{Y}) &\overset{ev_c}{\longrightarrow}& \mathbf{Y}(c) \\ (x, \phi) &\mapsto& \phi_c( id_c, x ) }

Then it remains to show that for every morphism of presheaves of the form X×AAfAY \mathbf{X} \times \mathbf{A} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{Y} there is a unique morphism f˜:A[X,Y]\widetilde f \;\colon\; \mathbf{A} \longrightarrow [\mathbf{X}, \mathbf{Y}] such that

(41)X×A X×f˜ X×[X,Y] f ev Y \array{ \mathbf{X} \times \mathbf{A} && \overset{ \mathbf{X} \times \widetilde f }{\longrightarrow} && \mathbf{X} \times [\mathbf{X}, \mathbf{Y}] \\ & {}_{\mathllap{ \mathrlap{f} }}\searrow && \swarrow_{ \mathrlap{ ev } } \\ && \mathbf{Y} }

The commutativity of this diagram means in components at c𝒞c \in \mathcal{C} that, that for all xX(c)x \in \mathbf{X}(c) and aA(c)a \in \mathbf{A}(c) we have

ev c(x,f˜ c(a)) (f˜ c(a)) c(id c,x) =f c(x,a) \begin{aligned} ev_c( x, \widetilde f_c(a) ) & \coloneqq (\widetilde f_c(a))_c( id_c, x ) \\ & = f_c( x, a ) \end{aligned}

Hence this fixes the component f˜ c(a) c\widetilde f_c(a)_c when its first argument is the identity morphism id cid_c. But let g:dcg \;\colon\; d \to c be any morphism and chase (id c,x)(id_c, x ) through the naturality diagram for f˜ c(a)\widetilde f_c(a):

Hom 𝒞(c,c)×X(c) (f˜ c(a)) c Y(c) g * g * Hom 𝒞(d,c)×X(d) (f˜ c(a)) d Y(d)AAAA{(id c,x)} {f c(x,a)} {(g,g *(x))} {f d(g *(x),g *(a))} \array{ Hom_{\mathcal{C}}(c,c) \times \mathbf{X}(c) &\overset{ (\widetilde f_c(a))_c }{\longrightarrow}& \mathbf{Y}(c) \\ {}^{\mathllap{ g^\ast }}\big\downarrow && \big\downarrow^{\mathrlap{ g^\ast }} \\ Hom_{\mathcal{C}}(d,c) \times \mathbf{X}(d) &\overset{ (\widetilde f_c(a))_d }{\longrightarrow}& \mathbf{Y}(d) } \phantom{AAAA} \array{ \{ (id_c, x ) \} &\longrightarrow& \{ f_c( x, a ) \} \\ \big\downarrow && \big\downarrow \\ \{ (g, g^\ast(x)) \} &\longrightarrow& \{ f_d( g^\ast(x), g^\ast(a) ) \} }

This shows that (f˜ c(a)) d(\widetilde f_c(a))_d is fixed to be given by

(42)(f˜ c(a)) d(g,x)=f d(x,g *(a)) (\widetilde f_c(a))_d( g, x' ) \;=\; f_d( x', g^\ast(a) )

at least on those pairs (g,x)(g,x') such that xx' is in the image of g *g^\ast.

But, finally, (f˜ c(a)) d(\widetilde f_c(a))_d is also natural in cc

A(c) f˜ c [X,Y](c) g * g * A(d) f˜ d [X,Y](d) \array{ \mathbf{A}(c) &\overset{ \widetilde f_c }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](c) \\ {}^{\mathllap{g^\ast}}\big\downarrow && \big\downarrow^{\mathrlap{g^\ast}} \\ \mathbf{A}(d) &\overset{ \widetilde f_d }{\longrightarrow}& [\mathbf{X},\mathbf{Y}](d) }

which implies that (42) must hold generally. Hence naturality implies that (41) indeed has a unique solution.

\,

The internal hom (Def. ) turns out to share all the abstract properties of the ordinary (external) hom-functor (Def. ), even though this is not completely manifest from its definition. We make this explicit by the following three propositions.

Proposition

(internal hom bifunctor)

For 𝒞\mathcal{C} a closed monoidal category (Def. ), there is a unique functor (Def. ) out of the product category (Def. ) of 𝒞\mathcal{C} with its opposite category (Def. )

[,]:𝒞 op×𝒞𝒞 [-,-] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}

such that for each X𝒞X \in \mathcal{C} it coincides with the internal hom [X,][X,-] (38) as a functor in the second variable, and such that there is a natural isomorphism

Hom(X,[Y,Z])Hom(XY,Z) Hom(X, [Y,Z]) \;\simeq\; Hom(X \otimes Y, Z)

which is natural not only in XX and ZZ, but also in YY.

Proof

We have a natural isomorphism for each fixed YY, and hence in particular for fixed YY and fixed ZZ by (38). With this the statement follows by Prop. .

In fact the 3-variable adjunction from Prop. even holds internally:

Proposition

(internal tensor/hom-adjunction)

In a symmetric closed monoidal category (def. ) there are natural isomorphisms

[XY,Z][X,[Y,Z]] [X \otimes Y, Z] \;\simeq\; [X, [Y,Z]]

whose image under Hom 𝒞(1,)Hom_{\mathcal{C}}(1,-) (see also Example below) are the defining natural bijections of Prop. .

Proof

Let A𝒞A \in \mathcal{C} be any object. By applying the natural bijections from Prop. , there are composite natural bijections

Hom 𝒞(A,[XY,Z]) Hom 𝒞(A(XY),Z) Hom 𝒞((AX)Y,Z) Hom 𝒞(AX,[Y,Z]) Hom 𝒞(A,[X,[Y,Z]]) \begin{aligned} Hom_{\mathcal{C}}(A , [X \otimes Y, Z]) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, [Y,Z]) \\ & \simeq Hom_{\mathcal{C}}(A, [X, [Y,Z]]) \end{aligned}

Since this holds for all AA, the fully faithfulness of the Yoneda embedding (Prop. ) says that there is an isomorphism [XY,Z][X,[Y,Z]][ X\otimes Y, Z ] \simeq [X, [Y,Z]]. Moreover, by taking A=1A = 1 in the above and using the left unitor isomorphisms A(XY)XYA \otimes (X \otimes Y) \simeq X \otimes Y and AXXA\otimes X \simeq X we get a commuting diagram

Hom 𝒞(1,[XY,Z)) Hom 𝒞(1,[X,[Y,Z]]) Hom 𝒞(XY,Z) Hom 𝒞(X,[Y,Z]). \array{ Hom_{\mathcal{C}}(1, [X\otimes Y, Z )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1, [X, [Y,Z]]) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, [Y,Z]) } \,.

Also the key respect of the hom-functor for limits is inherited by internal hom-functors

Proposition

(internal hom preserves limits)

Let 𝒞\mathcal{C} be a symmetric closed monoidal category with internal hom-bifunctor [,][-,-] (Prop. ). Then this bifunctor preserves limits in the second variable, and sends colimits in the first variable to limits:

[X,limj𝒥Y(j)]limj𝒥[X,Y(j)] [X, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Y(j)] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [X, Y(j)]

and

[limj𝒥Y(j),X]limj𝒥[Y(j),X] [\underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j),X] \;\simeq\; \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j),X]
Proof

For X𝒳X \in \mathcal{X} any object, [X,][X,-] is a right adjoint by definition, and hence preserves limits by Prop. .

For the other case, let Y:𝒞Y \;\colon\; \mathcal{L} \to \mathcal{C} be a diagram in 𝒞\mathcal{C}, and let C𝒞C \in \mathcal{C} be any object. Then there are isomorphisms

Hom 𝒞(C,limj𝒥Y(j),X) Hom 𝒞(Climj𝒥Y(j),X) Hom 𝒞(limj𝒥(CY(j)),X) limj𝒥Hom 𝒞((CY(j)),X) limj𝒥Hom 𝒞(C,[Y(j),X]) Hom 𝒞(C,limj𝒥[Y(j),X]) \begin{aligned} Hom_{\mathcal{C}}(C, \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) & \simeq Hom_{\mathcal{C}}( C \otimes \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X ) \\ & \simeq Hom_{\mathcal{C}}( \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( (C \otimes Y(j)), X ) \\ & \simeq \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} Hom_{\mathcal{C}}( C, [Y(j), X] ) \\ & \simeq Hom_{\mathcal{C}}( C, \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] ) \end{aligned}

which are natural in C𝒞C \in \mathcal{C}, where we used that the ordinary hom-functor preserves limits (Prop. ), and that the left adjoint C()C \otimes (-) preserves colimits, since left adjoints preserve colimits (Prop. ).

Hence by the fully faithfulness of the Yoneda embedding, there is an isomorphism

[limj𝒥Y(j),X]limj𝒥[Y(j),X]. \left[ \underset{\underset{j \in \mathcal{J}}{\longrightarrow}}{\lim} Y(j), X \right] \overset{\simeq}{\longrightarrow} \underset{\underset{j \in \mathcal{J}}{\longleftarrow}}{\lim} [Y(j), X] \,.

\,

Now that we have seen monoidal categories with various extra properties, we next look at functors which preserve these:

Definition

(monoidal functors)

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two monoidal categories (def. ). A lax monoidal functor between them is

  1. a functor (Def. )

    F:𝒞𝒟, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    (43)ϵ:1 𝒟F(1 𝒞) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a natural transformation (Def. )

    (44)μ x,y:F(x) 𝒟F(y)F(x 𝒞y) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,y𝒞x,y \in \mathcal{C}

satisfying the following conditions:

  1. (associativity) For all objects x,y,z𝒞x,y,z \in \mathcal{C} the following diagram commutes

    (F(x) 𝒟F(y)) 𝒟F(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟F(z)) μ x,yid idμ y,z F(x 𝒞y) 𝒟F(z) F(x) 𝒟(F(x 𝒞y)) μ x 𝒞y,z μ x,y 𝒞z F((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) F(x 𝒞(y 𝒞z)), \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,

    where a 𝒞a^{\mathcal{C}} and a 𝒟a^{\mathcal{D}} denote the associators of the monoidal categories;

  2. (unitality) For all x𝒞x \in \mathcal{C} the following diagrams commutes

    1 𝒟 𝒟F(x) ϵid F(1 𝒞) 𝒟F(x) F(x) 𝒟 μ 1 𝒞,x F(x) F( x 𝒞) F(1 𝒞x) \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }

    and

    F(x) 𝒟1 𝒟 idϵ F(x) 𝒟F(1 𝒞) r F(x) 𝒟 μ x,1 𝒞 F(x) F(r x 𝒞) F(x 𝒞1), \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,

    where 𝒞\ell^{\mathcal{C}}, 𝒟\ell^{\mathcal{D}}, r 𝒞r^{\mathcal{C}}, r 𝒟r^{\mathcal{D}} denote the left and right unitors of the two monoidal categories, respectively.

If ϵ\epsilon and alll μ x,y\mu_{x,y} are isomorphisms, then FF is called a strong monoidal functor.

If moreover (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are equipped with the structure of braided monoidal categories (def. ) with braidings τ 𝒞\tau^{\mathcal{C}} and τ 𝒟\tau^{\mathcal{D}}, respectively, then the lax monoidal functor FF is called a braided monoidal functor if in addition the following diagram commutes for all objects x,y𝒞x,y \in \mathcal{C}

F(x) 𝒞F(y) τ F(x),F(y) 𝒟 F(y) 𝒟F(x) μ x,y μ y,x F(x 𝒞y) F(τ x,y 𝒞) F(y 𝒞x). \array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.

A homomorphism f:(F 1,μ 1,ϵ 1)(F 2,μ 2,ϵ 2)f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2) between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation f x:F 1(x)F 2(x)f_x \;\colon\; F_1(x) \longrightarrow F_2(x) of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects x,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f(x) 𝒟f(y) F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f(x 𝒞y) F 2(x 𝒞y) \array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }

and

1 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f(1 𝒞) F 2(1 𝒞). \array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.

We write MonFun(𝒞,𝒟)MonFun(\mathcal{C},\mathcal{D}) for the resulting category of lax monoidal functors between monoidal categories 𝒞\mathcal{C} and 𝒟\mathcal{D}, similarly BraidMonFun(𝒞,𝒟)BraidMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between braided monoidal categories, and SymMonFun(𝒞,𝒟)SymMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between symmetric monoidal categories.

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.

If (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Proposition

For 𝒞F𝒟G\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E} two composable lax monoidal functors (def. ) between monoidal categories, then their composite FGF \circ G becomes a lax monoidal functor with structure morphisms

ϵ GF:1 ϵ GG(1 𝒟)G(ϵ F)G(F(1 𝒞)) \epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))

and

μ c 1,c 2 GF:G(F(c 1)) G(F(c 2))μ F(c 1),F(c 2) GG(F(c 1) 𝒟F(c 2))G(μ c 1,c 2 F)G(F(c 1 𝒞c 2)). \mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.

Algebras and modules

Definition

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (Def. ), then a monoid internal to (𝒞,,1)(\mathcal{C}, \otimes, 1) is

  1. an object A𝒞A \in \mathcal{C};

  2. a morphism e:1Ae \;\colon\; 1 \longrightarrow A (called the unit)

  3. a morphism μ:AAA\mu \;\colon\; A \otimes A \longrightarrow A (called the product);

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA μ A, \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,

    where aa is the associator isomorphism of 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ r A, \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,

    where \ell and rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (def. ) (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) with symmetric braiding τ\tau, then a monoid (A,μ,e)(A,\mu, e) as above is called a commutative monoid in (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) if in addition

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ A. \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.

A homomorphism of monoids (A 1,μ 1,e 1)(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1A 2 f \;\colon\; A_1 \longrightarrow A_2

in 𝒞\mathcal{C}, such that the following two diagrams commute

A 1A 1 ff A 2A 2 μ 1 μ 2 A 1 f A 2 \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }

and

1 𝒸 e 1 A 1 e 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(𝒞,,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in 𝒞\mathcal{C}, and CMon(𝒞,,1)CMon(\mathcal{C}, \otimes, 1) for its full subcategory of commutative monoids.

Example

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (Def. ), the tensor unit 11 is a monoid in 𝒞\mathcal{C} (def. ) with product given by either the left or right unitor

1=r 1:111. \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.

By lemma , these two morphisms coincide and define an associative product with unit the identity id:11id \colon 1 \to 1.

If (𝒞,,1)(\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.

Example

Given a symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given two commutative monoids (E i,μ i,e i)(E_i, \mu_i, e_i) i{1,2}i \in \{1,2\} (def. ), then the tensor product E 1E 2E_1 \otimes E_2 becomes itself a commutative monoid with unit morphism

e:111e 1e 2E 1E 2 e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2

(where the first isomorphism is, 1 1=r 1 1\ell_1^{-1} = r_1^{-1} (lemma )) and with product morphism given by

E 1E 2E 1E 2idτ E 2,E 1idE 1E 1E 2E 2μ 1μ 2E 1E 2 E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2

(where we are notationally suppressing the associators and where τ\tau denotes the braiding of 𝒞\mathcal{C}).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of (E i,μ i,e i)(E_i,\mu_i, e_i), and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Similarly one checks that for E 1=E 2=EE_1 = E_2 = E then the unit maps

EE1ideEE E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E
E1Ee1EE E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E

and the product map

μ:EEE \mu \;\colon\; E \otimes E \longrightarrow E

and the braiding

τ E,E:EEEE \tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E

are monoid homomorphisms, with EEE \otimes E equipped with the above monoid structure.

Definition

Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (A,μ,e)(A,\mu,e) a monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), then a left module object in (𝒞,,1)(\mathcal{C}, \otimes, 1) over (A,μ,e)(A,\mu,e) is

  1. an object N𝒞N \in \mathcal{C};

  2. a morphism ρ:ANN\rho \;\colon\; A \otimes N \longrightarrow N (called the action);

such that

  1. (unitality) the following diagram commutes:

    1N eid AN ρ N, \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,,

    where \ell is the left unitor isomorphism of 𝒞\mathcal{C}.

  2. (action property) the following diagram commutes

    (AA)N a A,A,N A(AN) Aρ AN μN ρ AN ρ N, \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,,

A homomorphism of left AA-module objects

(N 1,ρ 1)(N 2,ρ 2) (N_1, \rho_1) \longrightarrow (N_2, \rho_2)

is a morphism

f:N 1N 2 f\;\colon\; N_1 \longrightarrow N_2

in 𝒞\mathcal{C}, such that the following diagram commutes:

AN 1 Af AN 2 ρ 1 ρ 2 N 1 f N 2. \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,.

For the resulting category of modules of left AA-modules in 𝒞\mathcal{C} with AA-module homomorphisms between them, we write

AMod(𝒞). A Mod(\mathcal{C}) \,.
Example

Given a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ) with the tensor unit 11 regarded as a monoid in a monoidal category via example , then the left unitor

C:1CC \ell_C \;\colon\; 1\otimes C \longrightarrow C

makes every object C𝒞C \in \mathcal{C} into a left module, according to def. , over CC. The action property holds due to lemma . This gives an equivalence of categories

𝒞1Mod(𝒞) \mathcal{C} \simeq 1 Mod(\mathcal{C})

of 𝒞\mathcal{C} with the category of modules over its tensor unit.

Example

The archetypical case in which all these abstract concepts reduce to the basic familiar ones is the symmetric monoidal category Ab of abelian groups from example .

  1. A monoid in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a ring.

  2. A commutative monoid in in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently a commutative ring RR.

  3. An RR-module object in (Ab, ,)(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z}) (def. ) is equivalently an RR-module;

  4. The tensor product of RR-module objects (def. ) is the standard tensor product of modules.

  5. The category of module objects RMod(Ab)R Mod(Ab) (def. ) is the standard category of modules RModR Mod.

Example

Closely related to the example , but closer to the structure we will see below for spectra, are monoids in the category of chain complexes (Ch ,,)(Ch_\bullet, \otimes, \mathbb{Z}) from example . These monoids are equivalently differential graded algebras.

Proposition

In the situation of def. , the monoid (A,μ,e)(A,\mu, e) canonically becomes a left module over itself by setting ρμ\rho \coloneqq \mu. More generally, for C𝒞C \in \mathcal{C} any object, then ACA \otimes C naturally becomes a left AA-module by setting:

ρ:A(AC)a A,A,C 1(AA)CμidAC. \rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,.

The AA-modules of this form are called free modules.

The free functor FF constructing free AA-modules is left adjoint to the forgetful functor UU which sends a module (N,ρ)(N,\rho) to the underlying object U(N,ρ)NU(N,\rho) \coloneqq N.

AMod(𝒞)UF𝒞. A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,.
Proof

A homomorphism out of a free AA-module is a morphism in 𝒞\mathcal{C} of the form

f:ACN f \;\colon\; A\otimes C \longrightarrow N

fitting into the diagram (where we are notationally suppressing the associator)

AAC Af AN μid ρ AC f N. \array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,.

Consider the composite

f˜:C C1CeidACfN, \tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,,

i.e. the restriction of ff to the unit “in” AA. By definition, this fits into a commuting square of the form (where we are now notationally suppressing the associator and the unitor)

AC idf˜ AN ideid = AAC idf AN. \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,.

Pasting this square onto the top of the previous one yields

AC idf˜ AN ideid = AAC Af AN μid ρ AC f N, \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,,

where now the left vertical composite is the identity, by the unit law in AA. This shows that ff is uniquely determined by f˜\tilde f via the relation

f=ρ(id Af˜). f = \rho \circ (id_A \otimes \tilde f) \,.

This natural bijection between ff and f˜\tilde f establishes the adjunction.

Definition

Given a closed symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. , def. ), given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given (N 1,ρ 1)(N_1, \rho_1) and (N 2,ρ 2)(N_2, \rho_2) two left AA-module objects (def.), then

  1. the tensor product of modules N 1 AN 2N_1 \otimes_A N_2 is, if it exists, the coequalizer

    N 1AN 2AAAAρ 1(τ N 1,AN 2)N 1ρ 2N 1N 1coeqN 1 AN 2 N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2

    and if A()A \otimes (-) preserves these coequalizers, then this is equipped with the left AA-action induced from the left AA-action on N 1N_1

  2. the function module hom A(N 1,N 2)hom_A(N_1,N_2) is, if it exists, the equalizer

    hom A(N 1,N 2)equhom(N 1,N 2)AAAAAAhom(AN 1,ρ 2)(A())hom(ρ 1,N 2)hom(AN 1,N 2). hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,.

    equipped with the left AA-action that is induced by the left AA-action on N 2N_2 via

    Ahom(X,N 2)hom(X,N 2)Ahom(X,N 2)XidevAN 2ρ 2N 2. \frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,.

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8)

Proposition

Given a closed symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. , def. ), and given (A,μ,e)(A,\mu,e) a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ). If all coequalizers exist in 𝒞\mathcal{C}, then the tensor product of modules A\otimes_A from def. makes the category of modules AMod(𝒞)A Mod(\mathcal{C}) into a symmetric monoidal category, (AMod, A,A)(A Mod, \otimes_A, A) with tensor unit the object AA itself, regarded as an AA-module via prop. .

If moreover all equalizers exist, then this is a closed monoidal category (def. ) with internal hom given by the function modules hom Ahom_A of def. .

(e.g. Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8)

Proof sketch

The associators and braiding for A\otimes_{A} are induced directly from those of \otimes and the universal property of coequalizers. That AA is the tensor unit for A\otimes_{A} follows with the same kind of argument that we give in the proof of example below.

Example

For (A,μ,e)(A,\mu,e) a monoid (def. ) in a symmetric monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) (def. ), the tensor product of modules (def. ) of two free modules (def. ) AC 1A\otimes C_1 and AC 2A \otimes C_2 always exists and is the free module over the tensor product in 𝒞\mathcal{C} of the two generators:

(AC 1) A(AC 2)A(C 1C 2). (A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,.

Hence if 𝒞\mathcal{C} has all coequalizers, so that the category of modules is a monoidal category (AMod, A,A)(A Mod, \otimes_A, A) (prop. ) then the free module functor (def. ) is a strong monoidal functor (def. )

F:(𝒞,,1)(AMod, A,A). F \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (A Mod, \otimes_A, A) \,.
Proof

It is sufficient to show that the diagram

AAAAAAAidμμidAAμA A \otimes A \otimes A \underoverset {\underset{id \otimes \mu}{\longrightarrow}} {\overset{\mu \otimes id}{\longrightarrow}} {\phantom{AAAA}} A \otimes A \overset{\mu}{\longrightarrow} A

is a coequalizer diagram (we are notationally suppressing the associators), hence that A AAAA \otimes_A A \simeq A, hence that the claim holds for C 1=1C_1 = 1 and C 2=1C_2 = 1.

To that end, we check the universal property of the coequalizer:

First observe that μ\mu indeed coequalizes idμid \otimes \mu with μid\mu \otimes id, since this is just the associativity clause in def. . So for f:AAQf \colon A \otimes A \longrightarrow Q any other morphism with this property, we need to show that there is a unique morphism ϕ:AQ\phi \colon A \longrightarrow Q which makes this diagram commute:

AA μ A f ϕ Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}} \\ Q } \,.

We claim that

ϕ:Ar 1A1ideAAfQ, \phi \;\colon\; A \underoverset{\simeq}{r^{-1}}{\longrightarrow} A \otimes 1 \overset{id \otimes e}{\longrightarrow} A \otimes A \overset{f}{\longrightarrow} Q \,,

where the first morphism is the inverse of the right unitor of 𝒞\mathcal{C}.

First to see that this does make the required triangle commute, consider the following pasting composite of commuting diagrams

AA μ A idr 1 r 1 AA1 μid A1 ide ide AAA μid AA idμ f AA f Q. \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,.

Here the the top square is the naturality of the right unitor, the middle square commutes by the functoriality of the tensor product :𝒞×𝒞𝒞\otimes \;\colon\; \mathcal{C}\times \mathcal{C} \longrightarrow \mathcal{C} and the definition of the product category (Example ), while the commutativity of the bottom square is the assumption that ff coequalizes idμid \otimes \mu with μid\mu \otimes id.

Here the right vertical composite is ϕ\phi, while, by unitality of (A,μ,e)(A,\mu ,e), the left vertical composite is the identity on AA, Hence the diagram says that ϕμ=f\phi \circ \mu = f, which we needed to show.

It remains to see that ϕ\phi is the unique morphism with this property for given ff. For that let q:AQq \colon A \to Q be any other morphism with qμ=f q\circ \mu = f. Then consider the commuting diagram

A1 A ide = AA μ A f q Q, \array{ A \otimes 1 &\overset{\simeq}{\longleftarrow}& A \\ {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq} & \downarrow^{\mathrlap{=}} \\ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}} \\ Q } \,,

where the top left triangle is the unitality condition and the two isomorphisms are the right unitor and its inverse. The commutativity of this diagram says that q=ϕq = \phi.

Definition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) as in prop. , then a monoid (E,μ,e)(E, \mu, e) in (AMod, A,A)(A Mod , \otimes_A , A) (def. ) is called an AA-algebra.

Proposition

Given a monoidal category of modules (AMod, A,A)(A Mod , \otimes_A , A) in a monoidal category (𝒞,,1)(\mathcal{C},\otimes, 1) as in prop. , and an AA-algebra (E,μ,e)(E,\mu,e) (def. ), then there is an equivalence of categories

AAlg comm(𝒞)CMon(AMod)CMon(𝒞) A/ A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/}

between the category of commutative monoids in AModA Mod and the coslice category of commutative monoids in 𝒞\mathcal{C} under AA, hence between commutative AA-algebras in 𝒞\mathcal{C} and commutative monoids EE in 𝒞\mathcal{C} that are equipped with a homomorphism of monoids AEA \longrightarrow E.

(e.g. EKMM 97, VII lemma 1.3)

Proof

In one direction, consider a AA-algebra EE with unit e E:AEe_E \;\colon\; A \longrightarrow E and product μ E/A:E AEE\mu_{E/A} \colon E \otimes_A E \longrightarrow E. There is the underlying product μ E\mu_E

EAE AAA EE coeq E AE μ E μ E/A E. \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,.

By considering a diagram of such coequalizer diagrams with middle vertical morphism e Ee Ae_E\circ e_A, one find that this is a unit for μ E\mu_E and that (E,μ E,e Ee A)(E, \mu_E, e_E \circ e_A) is a commutative monoid in (𝒞,,1)(\mathcal{C}, \otimes, 1).

Then consider the two conditions on the unit e E:AEe_E \colon A \longrightarrow E. First of all this is an AA-module homomorphism, which means that

()AA ide E AE μ A ρ A e E E (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E }

commutes. Moreover it satisfies the unit property

A AE e Aid E AE μ E/A E. \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,.

By forgetting the tensor product over AA, the latter gives

AE eid EE A AE e Eid E AE μ E/A E = EAE e Eid EE ρ μ E E id E, \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,,

where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be pasted to the square ()(\star) above, to yield a commuting square

AA ide E AE e Eid EE μ A ρ μ E A e E E id E=AA e Ee E EE μ A μ E A e E E. \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,.

This shows that the unit e Ae_A is a homomorphism of monoids (A,μ A,e A)(E,μ E,e Ee A)(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A).

Now for the converse direction, assume that (A,μ A,e A)(A,\mu_A, e_A) and (E,μ E,e E)(E, \mu_E, e'_E) are two commutative monoids in (𝒞,,1)(\mathcal{C}, \otimes, 1) with e E:AEe_E \;\colon\; A \to E a monoid homomorphism. Then EE inherits a left AA-module structure by

ρ:AEe AidEEμ EE. \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,.

By commutativity and associativity it follows that μ E\mu_E coequalizes the two induced morphisms EAEAAEEE \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E. Hence the universal property of the coequalizer gives a factorization through some μ E/A:E AEE\mu_{E/A}\colon E \otimes_A E \longrightarrow E. This shows that (E,μ E/A,e E)(E, \mu_{E/A}, e_E) is a commutative AA-algebra.

Finally one checks that these two constructions are inverses to each other, up to isomorphism.

Definition

(lax monoidal functor)

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two monoidal categories (def. ). A lax monoidal functor between them is

  1. a functor

    F:𝒞𝒟, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    ϵ:1 𝒟F(1 𝒞) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a natural transformation

    μ x,y:F(x) 𝒟F(y)F(x 𝒞y) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,y𝒞x,y \in \mathcal{C}

satisfying the following conditions:

  1. (associativity) For all objects x,y,z𝒞x,y,z \in \mathcal{C} the following diagram commutes

    (F(x) 𝒟F(y)) 𝒟F(z) a F(x),F(y),F(z) 𝒟 F(x) 𝒟(F(y) 𝒟F(z)) μ x,yid idμ y,z F(x 𝒞y) 𝒟F(z) F(x) 𝒟(F(x 𝒞y)) μ x 𝒞y,z μ x,y 𝒞z F((x 𝒞y) 𝒞z) F(a x,y,z 𝒞) F(x 𝒞(y 𝒞z)), \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,,

    where a 𝒞a^{\mathcal{C}} and a 𝒟a^{\mathcal{D}} denote the associators of the monoidal categories;

  2. (unitality) For all x𝒞x \in \mathcal{C} the following diagrams commutes

    1 𝒟 𝒟F(x) ϵid F(1 𝒞) 𝒟F(x) F(x) 𝒟 μ 1 𝒞,x F(x) F( x 𝒞) F(1 𝒞x) \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }

    and

    F(x) 𝒟1 𝒟 idϵ F(x) 𝒟F(1 𝒞) r F(x) 𝒟 μ x,1 𝒞 F(x) F(r x 𝒞) F(x 𝒞1), \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,,

    where 𝒞\ell^{\mathcal{C}}, 𝒟\ell^{\mathcal{D}}, r 𝒞r^{\mathcal{C}}, r 𝒟r^{\mathcal{D}} denote the left and right unitors of the two monoidal categories, respectively.

If ϵ\epsilon and alll μ x,y\mu_{x,y} are isomorphisms, then FF is called a strong monoidal functor.

If moreover (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are equipped with the structure of braided monoidal categories (def. ) with braidings τ 𝒞\tau^{\mathcal{C}} and τ 𝒟\tau^{\mathcal{D}}, respectively, then the lax monoidal functor FF is called a braided monoidal functor if in addition the following diagram commutes for all objects x,y𝒞x,y \in \mathcal{C}

F(x) 𝒞F(y) τ F(x),F(y) 𝒟 F(y) 𝒟F(x) μ x,y μ y,x F(x 𝒞y) F(τ x,y 𝒞) F(y 𝒞x). \array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,.

A homomorphism f:(F 1,μ 1,ϵ 1)(F 2,μ 2,ϵ 2)f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2) between two (braided) lax monoidal functors is a monoidal natural transformation, in that it is a natural transformation f x:F 1(x)F 2(x)f_x \;\colon\; F_1(x) \longrightarrow F_2(x) of the underlying functors

compatible with the product and the unit in that the following diagrams commute for all objects x,y𝒞x,y \in \mathcal{C}:

F 1(x) 𝒟F 1(y) f(x) 𝒟f(y) F 2(x) 𝒟F 2(y) (μ 1) x,y (μ 2) x,y F 1(x 𝒞y) f(x 𝒞y) F 2(x 𝒞y) \array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} y) }

and

1 𝒟 ϵ 1 ϵ 2 F 1(1 𝒞) f(1 𝒞) F 2(1 𝒞). \array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,.

We write MonFun(𝒞,𝒟)MonFun(\mathcal{C},\mathcal{D}) for the resulting category of lax monoidal functors between monoidal categories 𝒞\mathcal{C} and 𝒟\mathcal{D}, similarly BraidMonFun(𝒞,𝒟)BraidMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between braided monoidal categories, and SymMonFun(𝒞,𝒟)SymMonFun(\mathcal{C},\mathcal{D}) for the category of braided monoidal functors between symmetric monoidal categories.

Remark

In the literature the term “monoidal functor” often refers by default to what in def. is called a strong monoidal functor. But for the purpose of the discussion of functors with smash product below, it is crucial to admit the generality of lax monoidal functors.

If (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) are symmetric monoidal categories (def. ) then a braided monoidal functor (def. ) between them is often called a symmetric monoidal functor.

Proposition

For 𝒞F𝒟G\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E} two composable lax monoidal functors (def. ) between monoidal categories, then their composite FGF \circ G becomes a lax monoidal functor with structure morphisms

ϵ GF:1 ϵ GG(1 𝒟)G(ϵ F)G(F(1 𝒞)) \epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}}))

and

μ c 1,c 2 GF:G(F(c 1)) G(F(c 2))μ F(c 1),F(c 2) GG(F(c 1) 𝒟F(c 2))G(μ c 1,c 2 F)G(F(c 1 𝒞c 2)). \mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,.
Proposition

(lax monoidal functors preserve monoids)

Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}}) be two monoidal categories (def. ) and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a lax monoidal functor (def. ) between them.

Then for (A,μ A,e A)(A,\mu_A,e_A) a monoid in 𝒞\mathcal{C} (def. ), its image F(A)𝒟F(A) \in \mathcal{D} becomes a monoid (F(A),μ F(A),e F(A))(F(A), \mu_{F(A)}, e_{F(A)}) by setting

μ F(A):F(A) 𝒞F(A)F(A 𝒞A)F(μ A)F(A) \mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)

(where the first morphism is the structure morphism of FF) and setting

e F(A):1 𝒟F(1 𝒞)F(e A)F(A) e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A)

(where again the first morphism is the corresponding structure morphism of FF).

This construction extends to a functor

Mon(F):Mon(𝒞, 𝒞,1 𝒞)Mon(𝒟, 𝒟,1 𝒟) Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})

from the category of monoids of 𝒞\mathcal{C} (def. ) to that of 𝒟\mathcal{D}.

Moreover, if 𝒞\mathcal{C} and 𝒟\mathcal{D} are symmetric monoidal categories (def. ) and FF is a braided monoidal functor (def. ) and AA is a commutative monoid (def. ) then so is F(A)F(A), and this construction extends to a functor between categories of commutative monoids:

CMon(F):CMon(𝒞, 𝒞,1 𝒞)CMon(𝒟, 𝒟,1 𝒟). CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,.
Proof

This follows immediately from combining the associativity and unitality (and symmetry) constraints of FF with those of AA.

\,

Enriched categories

The plain definition of categories in Def. is phrased in terms of sets. Via Example this assigns a special role to the category Set of all sets, as the “base” on top, or the “cosmos” inside which category theory takes place. For instance, the fact that hom-sets in a plain category are indeed sets, is what makes the hom-functor (Example ) take values in Set, and this, in turn, governs the form of the all-important Yoneda lemma (Prop. ) and Yoneda embedding (Prop. ) as statements about presheaves of sets (Example ).

At the same time, category theory witnesses the utility of abstracting away from concrete choices to their abstract properties that are actually used in constructions. This makes it natural to ask if one could replace the category Set by some other category 𝒱\mathcal{V} which could similarly serve as a “cosmos” inside which category theory may be developed.

Indeed, such 𝒱\mathcal{V}-enriched category theory (see Example below for the terminology) exists, beginning with the concept of 𝒱\mathcal{V}-enriched categories (Def. below) and from there directly paralleling, hence generalizing, plain category theory, as long as one assumes the “cosmos” category 𝒱\mathcal{V} to share a minimum of abstract properties with Set (Def. below).

This turns out to be most useful. In fact, the perspective of enriched categories is helpful already when 𝒱=\mathcal{V} = Set, in which case it reproduces plain category theory (Example below), for instance in that it puts the (co)limits of the special form of (co)ends (Def. below) to the forefront (discussed below).

\,

Definition

(cosmos)

A Bénabou cosmos for enriched category theory, or just cosmos, for short, is a symmetric (Def. ) closed monoidal category (Def. ) 𝒱\mathcal{V} which has all limits and colimits.

Example

(examples of cosmoi for enriched category theory)

The following are examples of cosmoi (Def. ):

  1. Sh(𝒞)Sh(\mathcal{C}) the sheaf topos (Def. ) over any site (Def. ) – by Prop. below.

    In particular:

    1. Set (Def. ) equipped with its cartesian closed category-structure (Example )

    2. sSet[Δ op,Set]\simeq [\Delta^{op}, Set] (Def. , Prop. )

  2. Grpd (Def. ) equipped with its cartesian closed category-structure (Example ).

  3. Cat (Def. ) equipped with its cartesian closed category-structure (Example ).

Example

underlying set of an object in a cosmos

Let 𝒱\mathcal{V} be a cosmos (Def. ), with 1𝒱1 \in \mathcal{V} its tensor unit (Def. ). Then the hom-functor (Def. ) out of 11

Hom 𝒱(1,):𝒱Set Hom_{\mathcal{V}}(1,-) \;\colon\; \mathcal{V} \longrightarrow Set

admits the structure of a lax monoidal functor (Def. ) to Set, with the latter regarded with its cartesian monoidal structure from Example .

Given V𝒱V \in \mathcal{V}, we call

Hom 𝒱(1,V)Set Hom_{\mathcal{V}}(1,V) \;\in\; Set

also the underlying set of VV.

Proof

Take the monoidal transformations (eq“MonoidalComponentsOfMonoidalFunctor) to be

Hom 𝒱(1,V 1)×Hom 𝒱(1,V 2)Hom 𝒱(1,V 1V 2) (1f 1V 1,1f 2V 2) (111f 1f 2V 1V 2) \array{ Hom_{\mathcal{V}}(1, V_1) \times Hom_{\mathcal{V}}(1, V_2) \longrightarrow Hom_{\mathcal{V}}(1, V_1 \otimes V_2) \\ \big( 1 \overset{f_1}{\to} V_1\;,\; 1 \overset{f_2}{\to} V_2 \big) &\mapsto& \big( 1 \overset{\simeq}{\to} 1 \otimes 1 \overset{f_1 \otimes f_2}{\longrightarrow} V_1 \otimes V_2 \big) }

and take the unit transformation (43)

* Hom 𝒱(1,1) \array{ \ast &\longrightarrow& Hom_{\mathcal{V}}(1,1) }

to pick id 1Hom 𝒱(1,1)id_1 \in Hom_{\mathcal{V}}(1,1).

Example

(underlying set of internal hom is hom-set)*

For 𝒱\mathcal{V} a cosmos (Def. ), let X,YObj 𝒱X,Y \in Obj_{\mathcal{V}} be two objects. Then the underlying set (Def. ) of their internal hom [X,Y]𝒱[X,Y] \in \mathcal{V} (Def. ) is the hom-set (Def. ):

ℋℴ𝓂 𝒱(1,[X,Y])Hom 𝒱(X,Y). \mathcal{Hom}_{\mathcal{V}}\left( 1, [X,Y]\right) \;\simeq\; Hom_{\mathcal{V}}(X,Y) \,.

This identification is the adjunction isomorphism (10) for the internal hom adjunction (38) followed composed with a unitor (Def. ).

Definition

(enriched category)

For 𝒱\mathcal{V} a cosmos (Def. ), a 𝒱\mathcal{V}-enriched category 𝒞\mathcal{C} is:

  1. a class Obj 𝒞Obj_{\mathcal{C}}, called the class of objects;

  2. for each a,bObj 𝒞a, b\in Obj_{\mathcal{C}}, an object

    𝒞(a,b)𝒱, \mathcal{C}(a,b)\in \mathcal{V} \,,

    called the 𝒱\mathcal{V}-object of morphisms between aa and bb;

  3. for each a,b,cObj(𝒞)a,b,c\in Obj(\mathcal{C}) a morphism in 𝒱\mathcal{V}

    a,b,c:𝒞(a,b)×𝒞(b,c)𝒞(a,c) \circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)

    out of the tensor product of hom-objects, called the composition operation;

  4. for each aObj(𝒞)a \in Obj(\mathcal{C}) a morphism Id a:*𝒞(a,a)Id_a \colon \ast \to \mathcal{C}(a,a), called the identity morphism on aa

such that the composition is associative and unital.

If the class Obj 𝒞Obj_{\mathcal{C}} happens to be a set (hence a small set instead of a proper class) then we say the 𝒱\mathcal{V}-enriched category 𝒞\mathcal{C} is small, as in Def. .

Example

(Set-enriched categories are plain categories)

An enriched category (Def. ) over the cosmos 𝒱=\mathcal{V} = Set, as in Example , is the same as a plain category (Def. ).

Example

(Cat-enriched categories are strict 2-categories)

An enriched category (Def. ) over the cosmos 𝒱=\mathcal{V} = Cat, as in Example , is the same as a strict 2-category (Def. ).

Example

(underlying category of an enriched category)

Let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ).

Using the lax monoidal structure (Def. ) on the hom functor (Example )

Hom 𝒱(1,):𝒱Set Hom_{\mathcal{V}}(1,-) \;\colon\; \mathcal{V} \longrightarrow Set

out of the tensor unit 1𝒞1 \in \mathcal{C} this induces a Set-enriched category |𝒞|\vert \mathcal{C}\vert with hence an ordinary category (Example ), with

  • Obj |𝒞|Obj 𝒞Obj_{\vert \mathcal{C}\vert} \;\coloneqq\; Obj_{\mathcal{C}};

  • Hom |𝒞|(X,Y)Hom 𝒱(1,𝒞(X,Y))Hom_{\vert \mathcal{C}\vert}( X, Y ) \;\coloneqq\; Hom_{\mathcal{V}}(1, \mathcal{C}(X,Y)).

It is in this sense that 𝒞\mathcal{C} is a plain category |𝒞|{\vert \mathcal{C}\vert} equipped with extra structure, and hence an “enriched category”.

The archetypical example is 𝒱\mathcal{V} itself:

Example

(𝒱\mathcal{V} as a 𝒱\mathcal{V}-enriched category)

Evert cosmos 𝒞\mathcal{C} (Def. ) canonically obtains the structure of a 𝒱\mathcal{V}-enriched category, def. :

the hom-objects are the internal homs

𝓋(X,Y)[X,Y] \mathcal{v}(X,Y) \coloneqq [X,Y]

and with composition

[X,Y]×[Y,Z][X,Z] [X,Y] \times [Y,Z] \longrightarrow [X,Z]

given by the adjunct under the (Cartesian product\dashv internal hom)-adjunction of the evaluation morphisms

X[XmY][Y,Z](ev,id)Y[Y,Z]evZ. X \otimes [XmY] \otimes [Y,Z] \overset{(ev, id)}{\longrightarrow} Y \otimes [Y,Z] \overset{ev}{\longrightarrow} Z \,.

The usual construction on categories, such as that of opposite categories (Def. ) and product categories (Def. ) have evident enriched analogs

Definition

(enriched opposite category and product category)

For 𝒱\mathcal{V} a cosmos, let 𝒞,𝒟\mathcal{C}, \mathcal{D} be 𝒱\mathcal{V}-enriched categories (Def. ).

  1. The opposite enriched category 𝒞 op\mathcal{C}^{op} is the enriched category with the same objects as 𝒞\mathcal{C}, with hom-objects

    𝒞 op(X,Y)𝒞(Y,X) \mathcal{C}^{op}(X,Y) \coloneqq \mathcal{C}(Y,X)

    and with composition given by braiding (37) followed by the composition in 𝒞\mathcal{C}:

    𝒞 op(X,Y)𝒞 op(Y,Z)=𝒞(Y,X)𝒞(Z,Y)τ𝒞(Z,Y)𝒞(Y,X) Z,Y,X𝒞(Z,X)=𝒞 op(X,Z). \mathcal{C}^{op}(X,Y) \otimes \mathcal{C}^{op}(Y,Z) = \mathcal{C}(Y,X) \otimes \mathcal{C}(Z,Y) \underoverset{\simeq}{\tau}{\longrightarrow} \mathcal{C}(Z,Y) \otimes \mathcal{C}(Y,X) \overset{\circ_{Z,Y,X}}{\longrightarrow} \mathcal{C}(Z,X) = \mathcal{C}^{op}(X,Z) \,.
  2. the enriched product category 𝒞×𝒟\mathcal{C} \times \mathcal{D} is the enriched category whose objects are pairs of objects (c,d)(c,d) with c𝒞c \in \mathcal{C} and d𝒟d\in \mathcal{D}, whose hom-spaces are the tensor product of the separate hom objects

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))𝒞(c 1,c 2)𝒟(d 1,d 2) (\mathcal{C}\times \mathcal{D})((c_1,d_1),\;(c_2,d_2) ) \coloneqq \mathcal{C}(c_1,c_2)\otimes \mathcal{D}(d_1,d_2)

    and whose composition operation is the braiding (37) followed by the tensor product of the separate composition operations:

    (𝒞×𝒟)((c 1,d 1),(c 2,d 2))(𝒞×𝒟)((c 2,d 2),(c 3,d 3)) = (𝒞(c 1,c 2)𝒟(d 1,d 2))(𝒞(c 2,c 3)𝒟(d 2,d 3)) τ (𝒞(c 1,c 2)𝒞(c 2,c 3))(𝒟(d 1,d 2)𝒟(d 2,d 3)) ( c 1,c 2,c 3)( d 1,d 2,d 3) 𝒞(c 1,c 3)𝒟(d 1,d 3) = (𝒞×𝒟)((c 1,d 1),(c 3,d 3)). \array{ (\mathcal{C}\times \mathcal{D})((c_1,d_1), \; (c_2,d_2)) \otimes (\mathcal{C}\times \mathcal{D})((c_2,d_2), \; (c_3,d_3)) \\ {}^{\mathllap{=}}\downarrow \\ \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{D}(d_1,d_2)\right) \otimes \left(\mathcal{C}(c_2,c_3) \otimes \mathcal{D}(d_2,d_3)\right) \\ \downarrow^{\mathrlap{\tau}}_{\mathrlap{\simeq}} \\ \left(\mathcal{C}(c_1,c_2) \otimes \mathcal{C}(c_2,c_3)\right) \otimes \left( \mathcal{D}(d_1,d_2) \otimes \mathcal{D}(d_2,d_3)\right) &\overset{ (\circ_{c_1,c_2,c_3}) \otimes (\circ_{d_1,d_2,d_3}) }{\longrightarrow} & \mathcal{C}(c_1,c_3) \otimes \mathcal{D}(d_1,d_3) \\ && \downarrow^{\mathrlap{=}} \\ && (\mathcal{C}\times \mathcal{D})((c_1,d_1),\; (c_3,d_3)) } \,.
Definition

(enriched functor)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories (Def. ).

A 𝒱\mathcal{V}-enriched functor from 𝒞\mathcal{C} to 𝒟\mathcal{D}

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

is

  1. a function

    F Obj:Obj 𝒞Obj 𝒟 F_{Obj} \;\colon\; Obj_{\mathcal{C}} \longrightarrow Obj_{\mathcal{D}}

    of objects;

  2. for each a,bObj 𝒞a,b \in Obj_{\mathcal{C}} a morphism in 𝒱\mathcal{V}

    F a,b:𝒞(a,b)𝒟(F 0(a),F 0(b)) F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))

    between hom-objects

such that this preserves composition and identity morphisms in the evident sense.

Example

(enriched hom-functor)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ). Then there is a 𝒱\mathcal{V}-enriched functor out of the enriched product category of 𝒞\mathcal{C} with its enriched opposite category (Def. )

𝒞(,):𝒞 op×𝒞𝒱 \mathcal{C}(-,-) \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}

to 𝒱\mathcal{V}, regarded as a 𝒱\mathcal{V}-enriched category (Example ), which sends a pair of objects X,Y𝒞X,Y \in \mathcal{C} to the hom-object 𝒞(X,Y)𝒱\mathcal{C}(X,Y) \in \mathcal{V}, and which acts on morphisms by composition in the evident way.

Example

(enriched presheaves)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ). Then a 𝒱\mathcal{V}-enriched functor (Def. )

F:𝒞𝒱 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{V}

to the archetypical 𝒱\mathcal{V}-enriched category from Example is:

  1. an object F aObj 𝒱F_a \in Obj_{\mathcal{V}} for each object aObj 𝒞a \in Obj_{\mathcal{C}};

  2. a morphism in 𝒱\mathcal{V} of the form

    F a𝒞(a,b)F b F_a \otimes \mathcal{C}(a,b) \longrightarrow F_b

    for all pairs of objects a,bObj(𝒞)a,b \in Obj(\mathcal{C})

    (this is the adjunct of F a,bF_{a,b} under the adjunction (38) on 𝒱\mathcal{V})

such that composition is respected, in the evident sense.

For every object c𝒞c \in \mathcal{C}, there is an enriched representable functor, denoted

y(c)𝒞(c,) y(c) \;\coloneqq\; \mathcal{C}(c,-)

(where on the right we have the enriched hom-functor from Example )

which sends objects to

y(c)(d)=𝒞(c,d)𝒱 y(c)(d) = \mathcal{C}(c,d) \in \mathcal{V}

and whose action on morphisms is, under the above identification, just the composition operation in 𝒞\mathcal{C}.

More generally, the following situation will be of interest:

Example

(enriched functor on enriched product category with opposite category)

An 𝒱\mathcal{V}-enriched functor (Def. ) into 𝒱\mathcal{V} (Example ) out of an enriched product category (Def. )

F:𝒞×𝒟𝒱 F \;\colon\; \mathcal{C} \times \mathcal{D} \longrightarrow \mathcal{V}

(an “enriched bifunctor”) has component morphisms of the form

F (c 1,d 1),(c 2,d 2):𝒞(c 1,c 2)𝒟(d 1,d 2)[F 0((c 1,d 1)),F 0((c 2,d 2))]. F_{(c_1,d_1),(c_2,d_2)} \;\colon\; \mathcal{C}(c_1,c_2) \otimes \mathcal{D}(d_1,d_2) \longrightarrow \big[ F_0((c_1,d_1)),F_0((c_2,d_2)) \big] \,.

By functoriality and under passing to adjuncts (Def. ) under (38) this is equivalent to two commuting actions

ρ c 1,c 2(d):𝒞(c 1,c 2)F 0((c 1,d))F 0((c 2,d)) \rho_{c_1,c_2}(d) \;\colon\; \mathcal{C}(c_1,c_2) \otimes F_0((c_1,d)) \longrightarrow F_0((c_2,d))

and

ρ d 1,d 2(c):𝒟(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \rho_{d_1,d_2}(c) \;\colon\; \mathcal{D}(d_1,d_2) \otimes F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.

In the special case of a functor out of the enriched product category of some 𝒱\mathcal{V}-enriched category 𝒞\mathcal{C} with its enriched opposite category (def. )

F:𝒞 op×𝒞𝒱 F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}

then this takes the form of a “pullback action” in the first variable

ρ c 2,c 1(d):𝒞(c 1,c 2)F 0((c 2,d))F 0((c 1,d)) \rho_{c_2,c_1}(d) \;\colon\; \mathcal{C}(c_1,c_2) \otimes F_0((c_2,d)) \longrightarrow F_0((c_1,d))

and a “pushforward action” in the second variable

ρ d 1,d 2(c):𝒞(d 1,d 2)F 0((c,d 1))F 0((c,d 2)). \rho_{d_1,d_2}(c) \;\colon\; \mathcal{C}(d_1,d_2) \otimes F_0((c,d_1)) \longrightarrow F_0((c,d_2)) \,.
Definition

(enriched natural transformation)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories (Def. ) and let

𝒞GF𝒟 \mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {} \mathcal{D}

be two 𝒱\mathcal{V}-enriched functors (Def. ) from 𝒞\mathcal{C} to 𝒟\mathcal{D}.

Then a 𝒱\mathcal{V}-enriched natural transformation

𝒞AAηAAGF𝒟 \mathcal{C} \underoverset {\underset{G}{\longrightarrow}} {\overset{F}{\longrightarrow}} {\phantom{AA}\Downarrow{\mathrlap{\eta}}\phantom{AA}} \mathcal{D}

is

  • for each cObj 𝒞c \in Obj_{\mathcal{C}} a choice of morphism

    η c:I𝒟(F(c),G(c)) \eta_c \;\colon\; I \longrightarrow \mathcal{D}(F(c),G(c))

    such that for each pair of objects c,d𝒞c,d \in \mathcal{C} the two morphisms (in 𝒱\mathcal{V})

(45)η dF():𝒞(c,d)r𝒞(c,d)IG c,dη c𝒟(G(c),G(d))𝒟(F(c),G(c)) F(c),G(c),G(d)𝒟(F(c),G(d)) \eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \overset{r}{\simeq} \mathcal{C}(c,d) \otimes I \overset{ G_{c,d} \otimes \eta_c }{\longrightarrow} \mathcal{D}(G(c),G(d)) \otimes \mathcal{D}( F(c), G(c) ) \overset{\circ_{F(c), G(c), G(d)}}{\longrightarrow} \mathcal{D}(F(c), G(d))

and

(46)G()η c:𝒞(c,d)I𝒞(c,d)η dF c,d𝒟(F(d),G(d))𝒟(F(c),F(d)) F(c),F(d),G(d)𝒟(F(c),G(d)) G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \overset{\ell}{\simeq} I \otimes \mathcal{C}(c,d) \overset{ \eta_d \otimes F_{c,d} }{\longrightarrow} \mathcal{D}(F(d), G(d)) \otimes \mathcal{D}(F(c), F(d)) \overset{\circ_{F(c),F(d), G(d)}}{\longrightarrow} \mathcal{D}(F(c), G(d))

agree.

Example

(functor category of enriched functors)

For 𝒱\mathcal{V} a cosmos (Def. ) let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories (Def. ). Then there is a category (Def. ) of enriched functors (Def. ), to be denoted

[𝒞,𝒟] [\mathcal{C}, \mathcal{D}]

whose objects are the enriched functors 𝒞F𝒟\mathcal{C} \overset{F}{\to} \mathcal{D} and whose morphisms are the enriched natural transformations between these (Def. ).

In the case that 𝒱=\mathcal{V} = Set, via Def. , with SetSet-enriched categories identified with plain categories via Example , this coincides with the functor category from Example .

Notice that, at this point, [𝒞,𝒟][\mathcal{C}, \mathcal{D}] is a plain category, not itself a 𝒱\mathcal{V}-enriched category, unless 𝒱=\mathcal{V} = Set. But it may be enhanced to one, this is Def. below.

There is now the following evident generalization of the concept of adjoint functors (Def. ) from plain category theory to enriched category theory:

Definition

(enriched adjunction)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories (Def. ). Then an adjoint pair of 𝒱\mathcal{V}-enriched functors or enriched adjunction

𝒞RL𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

is a pair of 𝒱\mathcal{V}-enriched functors (Def. ), as shown, such that there is a 𝒱\mathcal{V}-enriched natural isomorphism (Def. ) between enriched hom-functors (Def. ) of the form

(47)𝒞(L(),)𝒟(,R()). \mathcal{C}(L(-),-) \;\simeq\; \mathcal{D}(-,R(-)) \,.
Definition

(enriched equivalence of categories)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two 𝒱\mathcal{V}-enriched categories (Def. ). Then an equivalence of enriched categories

𝒞AAAARL𝒟 \array{ \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\phantom{AA} \simeq \phantom{AA}} \mathcal{D} }

is a pair of 𝒱\mathcal{V}-enriched functors back and forth, as shown (Def. ), together with 𝒱\mathcal{V}-enriched natural isomorphisms (Def. ) between their composition and the identity functors:

id 𝒟RLAAAandAAALRid 𝒞. id_{\mathcal{D}} \overset{\simeq}{\Rightarrow} R \circ L \phantom{AAA} \text{and} \phantom{AAA} L \circ R \overset{\simeq}{\Rightarrow} id_{\mathcal{C}} \,.

\,

Universal constructions

What makes category theory be theory, as opposed to just a language, is the concept of universal constructions. This refers to the idea of objects with a prescribed property which are universal with this property, in that they “know about” or “subsume” every other object with that same kind of property. Category theory allows to make precise what this means, and then to discover and prove theorems about it.

Universal constructions are all over the place in mathematics. Iteratively finding the universal constructions in a prescribed situation essentially amounts to systematically following the unravelling of the given situation or problem or theory that one is studying.

There are several different formulations of the concept of universal constructions, discussed below:

But these three kinds of constructions all turn out to be special cases of each other, hence they really reflect different perspectives on a single topic of universal constructions. In fact, all three are also special cases of the concept of adjunction (Def. ), thus re-amplifying that category theory is really the theory of adjunctions and hence, if we follow (Lambek 82), of duality.

\,

Limits and colimits

Maybe the most hands-on version of universal constructions are limits (Def. below), which is short for limiting cones (Remark below). The formally dual concept (Example ) is called colimits (which are hence limits in an opposite category). Other terminology is in use, too:

A\phantom{A} lim\underset{\longleftarrow}{\lim} A\phantom{A}A\phantom{A} lim\underset{\longrightarrow}{\lim} A\phantom{A}
A\phantom{A} limit A\phantom{A}A\phantom{A} colimit A\phantom{A}
A\phantom{A} inverse limit A\phantom{A}A\phantom{A} direct limitA\phantom{A}

There is a variety of different kinds of limits/colimits, depending on the diagram shape that they are limiting (co-)cones over. This includes universal constructions known as equalizers, products, fiber products/pullbacks, filtered limits and various others, all of which are basic tools frequently used whenever category theory applies.

A key fact of category theory, regarding limits, is that right adjoints preserve limits and left adjoints preserve colimits (Prop. below). This will be used all the time. A partial converse to this statement is that if a functor preserves limits/colimits, then its adjoint functor is, if it exists, objectwise given by a limit/colimit over a comma category under/over the given functor (Prop. below). Since these comma categories are in general not small, this involves set-theoretic size subtleties that are dealt with by the adjoint functor theorem (Remark below). We discuss in detail a very special but also very useful special case of this in Prop. , further below.

\,

Definition

(limit and colimit)

Let 𝒞\mathcal{C} be a small category (Def. ), and let 𝒟\mathcal{D} be any category (Def. ). In this case one also says that a functor

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

is a diagram of shape 𝒞\mathcal{C} in 𝒟\mathcal{D}.

Recalling the functor category (Example ) [𝒞,𝒟][\mathcal{C}, \mathcal{D}], there is the constant diagram-functor

const:𝒟[𝒞,𝒟] const \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}, \mathcal{D}]

which sends an object X𝒟X \in \mathcal{D} to the functor that sends every c𝒞c \in \mathcal{C} to XX, and every morphism in 𝒞\mathcal{C} to the identity morphism on XX. Accordingly, every morphism in 𝒟\mathcal{D} is sent by constconst to the natural transformation (Def. ) all whose components are equal to that morphism.

Now:

  1. if constconst has a right adjoint (Def. ), this is called the construction of forming the limiting cone of 𝒞\mathcal{C}-shaped diagrams in 𝒟\mathcal{D}, or just limit (or inverse limit) for short, and denoted

    lim𝒞:[𝒞,𝒟]𝒟 \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \;\colon\; [\mathcal{C}, \mathcal{D}] \longrightarrow \mathcal{D}
  2. if constconst has a left adjoint (Def. ), this is called the construction of forming the colimiting cocone of 𝒞\mathcal{C}-shaped diagrams in 𝒟\mathcal{D}, or just colimit (or direct limit) for short, and denoted

    lim𝒞:[𝒞,𝒟]𝒟 \underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim} \;\colon\; [\mathcal{C}, \mathcal{D}] \longrightarrow \mathcal{D}
(48)[𝒞,𝒟]AAlim𝒞AA AAconstAA AAlim𝒞AA𝒟. [\mathcal{C}, \mathcal{D}] \array{ \overset{ \phantom{AA}\underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim} \phantom{AA} }{\longrightarrow} \\ \overset{\phantom{AA} const \phantom{AA} }{ \longleftarrow } \\ \overset{ \phantom{AA} \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \phantom{AA}}{\longrightarrow} } \mathcal{D} \,.

If lim𝒞\underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} (lim𝒞\underset{\underset{\mathcal{C}}{\longrightarrow}}{\lim}) exists for a given 𝒟\mathcal{D}, one says that 𝒟\mathcal{D} has all limits (_has all colimits_) of shape 𝒞\mathcal{C}_ or that all limits (colimits) of shape 𝒟\mathcal{D} exist in 𝒟\mathcal{D}. If this is the case for all small diagrams 𝒞\mathcal{C}, one says that 𝒟\mathcal{D} has all limits (_has all colimits_) or that all limits exist in 𝒟\mathcal{D}, (_all colimits exist in 𝒟\mathcal{D}.)

Remark

(limit cones)

Unwinding Definition of limits and colimits, it says the following.

First of all, for d𝒟d \in \mathcal{D} any object and F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} any functor, a natural transformation (Def. ) of the form

(49)const diF const_d \overset{i}{\Rightarrow} F

has component morphisms

d i c F(c) \array{ d \\ \big\downarrow^{\mathrlap{i_c}} \\ F(c) }

in 𝒟\mathcal{D}, for each c𝒞c \in \mathcal{C}, and the naturality condition (4) says that these form a commuting diagram (Def. ) of the form

(50) d i c 1 i c 2 F(c 1) AAF(f)AA F(c 2) \array{ && d \\ & {}^{\mathllap{ i_{c_1} } }\swarrow && \searrow^{\mathrlap{ i_{c_2} }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} && F(c_2) }

for each morphism c 1fc 2c_1 \overset{f}{\to} c_2 in 𝒞\mathcal{C}. Due to the look of this diagram, one also calls such a natural transformation a cone over the functor FF.

Now the counit (Def. ) of the (constlim)(const \dashv \underset{\longleftarrow}{\lim})-adjunction (48) is a natural transformation of the form

const limFAAϵ FAAF const_{\underset{\longleftarrow}{\lim} F} \overset{ \phantom{AA} \epsilon_{F} \phantom{AA} }{\longrightarrow} F

and hence is, in components, a cone (50) over FF:

(51) limF ϵ F(c 1) ϵ F(c 2) F(c 1) AAF(f)AA F(c 2) \array{ && \underset{\longleftarrow}{\lim} F \\ & {}^{\mathllap{ \epsilon_F(c_1) } }\swarrow && \searrow^{\mathrlap{ \epsilon_F(c_2) }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} && F(c_2) }

to be called the limiting cone over FF

But the universal property of adjunctions says that this is a very special cone: By Prop. the defining property of the limit is equivalently that for every natural transformation of the form (49), hence for every cone of the form (50), there is a unique natural transformation

const d i˜ const lim \array{ const_d &\overset{\widetilde i}{\Rightarrow}& const_{ \underset{ \longleftarrow }{\lim} } }

which, due to constancy of the two functors applied in the naturality condition (4), has a constant component morphism

(52)di˜limF d \overset{ \widetilde i }{\longrightarrow} \underset{\longleftarrow}{\lim} F

such that

const d i˜ const limF ϵ F i F \array{ const_d && \overset{\widetilde i}{\longrightarrow} && const_{ \underset{\longleftarrow}{\lim} F } \\ & {}_{\mathllap{ \epsilon_F }} \searrow && \swarrow_{ \mathrlap{i} } \\ && F }

hence such that (52) factors the given cone (50) through the special cone (51):

d i c 1 i c 2 F(c 1) AAF(f)AAF(c 2)AAA=AAA d i˜ limF ϵ F(c 1) ϵ F(c 2) F(c 1) AAF(f)AAF(c 2) \array{ && d \\ & {}^{\mathllap{ i_{c_1} } }\swarrow && \searrow^{\mathrlap{ i_{c_2} }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} F(c_2) } \phantom{AAA} = \phantom{AAA} \array{ && d \\ && \big\downarrow^{ \mathrlap{ \widetilde i } } \\ && \underset{\longleftarrow}{\lim} F \\ & {}^{\mathllap{ \epsilon_F(c_1) } }\swarrow && \searrow^{\mathrlap{ \epsilon_F(c_2) }} \\ F(c_1) && \underset{ \phantom{AA} F(f) \phantom{AA} }{\longrightarrow} F(c_2) }

In this case one also says that i˜\widetilde i is a morphism of cones.

Hence a limit cone is a cone over FF, such that every other cone factors through it in a unique way.

Of course this concept of (co)limiting cone over a functor F:𝒞𝒟F \;\colon\; \mathcal{C} \to \mathcal{D} makes sense also when

  1. 𝒞\mathcal{C} is not small,

  2. and/or when a (co-)limiting cone exists only for some but not for all functors of this form.

Example

(terminal/initial object is empty limit/colimit)

Let 𝒞\mathcal{C} be a category, and let *𝒞\ast \in \mathcal{C} be an object. The following are equivalent:

  1. *\ast is a terminal object of 𝒞\mathcal{C} (Def. );

  2. *\ast is the limit of the empty diagram.

And formally dual (example ): Let 𝒞\emptyset \in \mathcal{C} be an object. The following are equivalent:

  1. \emptyset is an initial object of 𝒞\mathcal{C} (Def. );

  2. \emptyset is the colimit of the empty diagram.

Proof

We discuss the case of the terminal object, the other case is formally dual (Example ).

It suffices to observe that a cone over the empty diagram (Remark ) is clearly just a plain object of 𝒞\mathcal{C}. Hence a morphism of such cones is just a plain morphism of 𝒞\mathcal{C}. This way the condition on a limiting cone is now manifestly the same as the condition on a terminal object.

Example

(initial object is limit over identity functor)

Let 𝒞\mathcal{C} be a category, and let 𝒞\emptyset \in \mathcal{C} be an object. The following are equivalent:

  1. \emptyset is an initial object of 𝒞\mathcal{C} (Def. );

  2. \emptyset is the tip of a limit cone (Remark ) over the identity functor on 𝒞\mathcal{C}.

Proof

First let \emptyset be an initial object. Then, by definition, it is the tip of a unique cone over the identity functor

(53)const AA i i c 1 i c 2 id 𝒞 c 1 f c 2 \array{ const_{\emptyset}&\phantom{AA}& && \emptyset \\ {}^{\mathllap{i^{\emptyset}}}\Downarrow && & {}^{\mathllap{i^{\emptyset}_{c_1}}}\swarrow && \searrow^{\mathrlap{i^{\emptyset}_{c_2}}} \\ id_{\mathcal{C}} && c_1 && \underset{f}{\longrightarrow} && c_2 }

We need to show that that every other cone i xi^x

const x AA x i x i c 1 x i c 2 x id 𝒞 c 1 f c 2 \array{ const_{x}&\phantom{AA}& && x \\ {}^{\mathllap{\mathllap{i^x}}}\Downarrow && & {}^{i^x_{c_1}}\swarrow && \searrow^{\mathrlap{i^x_{c_2}}} \\ id_{\mathcal{C}} && c_1 && \underset{f}{\longrightarrow} && c_2 }

factors uniquely through i i^\emptyset.

First of all, since the cones are over the identity functor, there is the component i x:xi^x_{\emptyset} \;\colon\; x \to \emptyset, and it is a morphism of cones.

To see that this is the unique morphism of cones, consider any morphism of cones j xj^x_\emptyset, hence a morphism in 𝒞\mathcal{C} such that i c x=i c j xi^x_c = i^\emptyset_c \circ j^x_\emptyset for all c𝒞c \in \mathcal{C}. Taking here c=c = \emptyset yields

i x =i =id j x =j x, \begin{aligned} i^x_\emptyset & = \underset{ = id_\emptyset }{\underbrace{i^\emptyset_{\emptyset}}} \circ j^x_{\emptyset} \\ & = j^x_\emptyset \,, \end{aligned}

where under the brace we used that \emptyset is initial. This proves that i i^\emptyset is the limiting cone.

For the converse, assume now that i i^\emptyset is a limiting cone over the identity functor, with labels as in (53). We need to show that its tip \emptyset is an initial object.

Now the cone condition applied for any object x𝒞x \in \mathcal{C} over the morphims fi x f \coloneqq i^\emptyset_x says that

i x i =i x i^{\emptyset}_x \circ i^\emptyset_\emptyset = i^\emptyset_x

which means that i i^\emptyset_\emptyset constitutes a morphism of cones from i i^\emptyset to itself. But since i i^\emptyset is assumed to be a limiting cone, and since the identity morphism on \emptyset is of course also a morphism of cones from i i^\emptyset to itsely, we deduce that

(54)i =id . i^\emptyset_\emptyset \;=\; id_{\emptyset} \,.

Now consider any morphism of the form fx\emptyset \overset{f}{\to} x. Since we already have the morphism i x x\emptyset \overset{i^\emptyset_x}{\to} x, to show initiality of \emptyset we need to show that f=i x f = i^\emptyset_x.

Indeed, the cone condition of i x i^\emptyset_x applied to ff now yields

i x =fi =id =f, \begin{aligned} i^\emptyset_x & = f \circ \underset{ = id_{\emptyset} }{\underbrace{i^\emptyset_\emptyset}} \\ & = f\,, \end{aligned}

where under the brace we used (54).

Example

(limits of presheaves are computed objectwise)

Let 𝒞\mathcal{C} be a category and write [𝒞 op,Set][\mathcal{C}^{op}, Set] for its category of presheaves (Example ). Let moreover 𝒟\mathcal{D} be a small category and consider any functor

F:𝒟[𝒞 op,𝒟], F \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}^{op}, \mathcal{D}] \,,

hence a 𝒟\mathcal{D}-shaped diagram in the category of presheaves.

Then

  1. The limit (Def. ) of FF exists, and is the presheaf which over any object c𝒞c \in \mathcal{C} is given by the limit in Set of the values of the presheaves at cc:

    (limd𝒟F(d))(c)limd𝒟F(d)(c) \left( \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d)(c)
  2. The colimit (Def. ) of FF exists, and is the presheaf which over any object c𝒞c \in \mathcal{C} is given by the colimit in Set of the values of the presheaves at cc:

    (limd𝒟F(d))(c)limd𝒟F(d)(c) \left( \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d)(c)
Proof

We discuss the case of limits, the other case is formally dual (Example ).

Observe that there is a canonical equivalence (Def. )

[𝒟,[𝒞 op,Set]][𝒟×𝒞 op,Set] [\mathcal{D}, [\mathcal{C}^{op}, \Set]] \simeq [\mathcal{D} \times \mathcal{C}^{op}, Set]

where 𝒟×𝒞 op\mathcal{D} \times \mathcal{C}^{op} is the product category.

This makes manifest that a functor F:𝒟[𝒞 op,Set]F \;\colon\; \mathcal{D} \to [\mathcal{C}^{op}, Set] is equivalently a diagram of the form

F(d 1)(c 2) AA F(d 2)(c 2) F(d 1)(c 1) AA F(d 2)(c 1) \array{ && \vdots && && \vdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ && \vdots && && \vdots }

Then observe that taking the limit of each “horizontal row” in such a diagram indead does yield a presheaf on 𝒞\mathcal{C}, in that the construction extends from objects to morphisms, and uniquely so: This is because for any morphism c 1gc 2c_1 \overset{g}{\to} c_2 in 𝒞\mathcal{C}, a cone over F()(c 2)F(-)(c_2) (Remark ) induces a cone over F()(c 1)F(-)(c_1), by vertical composition with F()(g)F(-)(g)

limd𝒟F(d)(c 2) F(d 1)(c 2) AA F(d 2)(c 2) F(d 1)(g) F(d 2)(g) F(d 1)(c 1) AA F(d 2)(c 1) \array{ && \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ & {}^{ }\swarrow && \searrow \\ F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) \\ {}^{\mathllap{F(d_1)(g)}}\big\downarrow && && \big\downarrow^{\mathrlap{F(d_2)(g)}} \\ F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) }

From this, the universal property of limits of sets (as in Remark ) implies that there is a unique morphism between the pointwise limits which constitutes a presheaf over 𝒞\mathcal{C}

limd𝒟F(d)(c 2) limd𝒟F(d)(g) limd𝒟F(d)(c 1) \array{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow^{\mathrlap{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(g) }} \\ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) }

and that is the tip of a cone over the diagram F()F(-) in presheaves.

Hence it remains to see that this cone of presheaves is indeed universal.

Now if II is any other cone over FF in the category of presheaves, then by the universal property of the pointswise limits, there is for each c𝒞c \in \mathcal{C} a unique morphism of cones in sets

I(c)limd𝒟F(d)(c). I(c) \longrightarrow \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c) \,.

Hence there is at most one morphisms of cones of presheaves, namely if these components make all their naturality squares commute.

I(c 2) limd𝒟F(d)(c 2) I(c 1) limd𝒟F(d)(c 1). \array{ I(c_2) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow && \big\downarrow \\ I(c_1) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) } \,.

But since everything else commutes, the two ways of going around this diagram constitute two morphisms from a cone over F()(c 1)F(-)(c_1) to the limit cone over F()(c 1)F(-)(c_1), and hence they must be equal, by the universal property of limits.

Proposition

(hom-functor preserves limits)

Let 𝒞\mathcal{C} be a category and write

Hom 𝒞:𝒞 op×𝒞Set Hom_{\mathcal{C}} \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow Set

for its hom-functor. This preserves limits (Def. ) in both its arguments (recalling that a limit in the opposite category 𝒞 op\mathcal{C}^{op} is a colimit in 𝒞\mathcal{C}).

More in detail, let X :𝒞X_\bullet \colon \mathcal{I} \longrightarrow \mathcal{C} be a diagram. Then:

  1. If the limit lim iX i\underset{\longleftarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y𝒞Y \in \mathcal{C} there is a natural isomorphism

    Hom 𝒞(Y,lim iX i)lim i(Hom 𝒞(Y,X i)), Hom_{\mathcal{C}}\left(Y, \underset{\longleftarrow}{\lim}_i X_i \right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( Y, X_i \right) \right) \,,

    where on the right we have the limit over the diagram of hom-sets given by

    Hom 𝒞(Y,)X:X𝒞Hom 𝒞(Y,)Set. Hom_{\mathcal{C}}(Y,-) \circ X \;\colon\; \mathcal{I} \overset{X}{\longrightarrow} \mathcal{C} \overset{Hom_{\mathcal{C}}(Y,-) }{\longrightarrow} Set\,.
  2. If the colimit lim iX i\underset{\longrightarrow}{\lim}_i X_i exists in 𝒞\mathcal{C} then for all Y𝒞Y \in \mathcal{C} there is a natural isomorphism

    Hom 𝒞(lim iX i,Y)lim i(Hom 𝒞(X i,Y)), Hom_{\mathcal{C}}\left(\underset{\longrightarrow}{\lim}_i X_i ,Y\right) \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{\mathcal{C}}\left( X_i , Y\right) \right) \,,

    where on the right we have the limit over the diagram of hom-sets given by

    Hom 𝒞(,Y)X: opX𝒞 opHom 𝒞(,Y)Set. Hom_{\mathcal{C}}(-,Y) \circ X \;\colon\; \mathcal{I}^{op} \overset{X}{\longrightarrow} \mathcal{C}^{op} \overset{Hom_{\mathcal{C}}(-,Y) }{\longrightarrow} Set\,.
Proof

We give the proof of the first statement, the proof of the second statement is formally dual (Example ).

First observe that, by the very definition of limiting cones, maps out of some YY into them are in natural bijection with the set Cones(Y,X )Cones\left(Y, X_\bullet \right) of cones over the diagram X X_\bullet with tip YY:

Hom(Y,lim iX i)Cones(Y,X ). Hom\left( Y, \underset{\longleftarrow}{\lim}_{i} X_i \right) \;\simeq\; Cones\left( Y, X_\bullet \right) \,.

Hence it remains to show that there is also a natural bijection like so:

Cones(Y,X )lim i(Hom(Y,X i)). Cones\left( Y, X_\bullet \right) \;\simeq\; \underset{\longleftarrow}{\lim}_{i} \left( Hom(Y,X_i) \right) \,.

Now, again by the very definition of limiting cones, a single element in the limit on the right is equivalently a cone of the form

{ * const p i const p j Hom(Y,X i) X α() Hom(Y,X j)} i,jObj(),αHom (i,j). \left\{ \array{ && \ast \\ & {}^{\mathllap{const_{p_i}}}\swarrow && \searrow^{\mathrlap{const_{p_j}}} \\ Hom(Y,X_i) && \underset{X_\alpha \circ (-)}{\longrightarrow} && Hom(Y,X_j) } \right\}_{i, j \in Obj(\mathcal{I}), \alpha \in Hom_{\mathcal{I}}(i,j) } \,.

This is equivalently for each object ii \in \mathcal{I} a choice of morphism p i:YX ip_i \colon Y \to X_i , such that for each pair of objects i,ji,j \in \mathcal{I} and each αHom (i,j)\alpha \in Hom_{\mathcal{I}}(i,j) we have X αp i=p jX_\alpha \circ p_i = p_j. And indeed, this is precisely the characterization of an element in the set Cones(Y,X }) Cones\left( Y, X_\bullet\} \right).

Example

(initial and terminal object in terms of adjunction)

Let 𝒞\mathcal{C} be a category (Def. ).

  1. The following are equivalent:

    1. 𝒞\mathcal{C} has a terminal object (Def. );

    2. the unique functor 𝒞*\mathcal{C} \to \ast (Def. ) to the terminal category (Example ) has a right adjoint (Def. )

      *𝒞 \ast \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}

    Under this equivalence, the terminal object is identified with the image under the right adjoint of the unique object of the terminal category.

  2. Dually, the following are equivalent:

    1. 𝒞\mathcal{C} has an initial object (Def. );

    2. the unique functor 𝒞*\mathcal{C} \to \ast to the terminal category has a left adjoint

      𝒞* \mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {\bot} \ast

    Under this equivalence, the initial object is identified with the image under the left adjoint of the unique object of the terminal category.

Proof

Since the unique hom-set in the terminal category is the singleton, the hom-isomorphism (10) characterizing the adjoint functors is directly the universal property of an initial object in 𝒞\mathcal{C}

Hom 𝒞(L(*),X)Hom *(*,R(X))=* Hom_{\mathcal{C}}( L(\ast) , X ) \;\simeq\; Hom_{\ast}( \ast, R(X) ) = \ast

or of a terminal object

Hom 𝒞(X,R(*))Hom *(L(X),*)=*, Hom_{\mathcal{C}}( X , R(\ast) ) \;\simeq\; Hom_{\ast}( L(X), \ast ) = \ast \,,

respectively.

Proposition

(left adjoints preserve colimits and right adjoints preserve limits)

Let (LR):𝒟𝒞(L \dashv R) \colon \mathcal{D} \to \mathcal{C} be a pair of adjoint functors (Def. ). Then

  • LL preserves all colimits (Def. ) that exist in 𝒞\mathcal{C},

  • RR preserves all limits (Def. ) in 𝒟\mathcal{D}.

Proof

Let y:I𝒟y : I \to \mathcal{D} be a diagram whose limit lim iy i\lim_{\leftarrow_i} y_i exists. Then we have a sequence of natural isomorphisms, natural in xCx \in C

Hom 𝒞(x,Rlim iy i) Hom 𝒟(Lx,lim iy i) lim iHom 𝒟(Lx,y i) lim iHom 𝒞(x,Ry i) Hom 𝒞(x,lim iRy i), \begin{aligned} Hom_{\mathcal{C}}(x, R {\lim_\leftarrow}_i y_i) & \simeq Hom_{\mathcal{D}}(L x, {\lim_\leftarrow}_i y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{D}}(L x, y_i) \\ & \simeq {\lim_\leftarrow}_i Hom_{\mathcal{C}}( x, R y_i) \\ & \simeq Hom_{\mathcal{C}}( x, {\lim_\leftarrow}_i R y_i) \,, \end{aligned}

where we used the hom-isomorphism (10) and the fact that any hom-functor preserves limits (Def. ). Because this is natural in xx the Yoneda lemma implies that we have an isomorphism

Rlim iy ilim iRy i. R {\lim_\leftarrow}_i y_i \simeq {\lim_\leftarrow}_i R y_i \,.

The argument that shows the preservation of colimits by LL is analogous.

Proposition

(limits commute with limits)

Let 𝒟\mathcal{D} and 𝒟\mathcal{D}' be small categories (Def. ) and let 𝒞\mathcal{C} be a category (Def. ) which admits limits (Def. ) of shape 𝒟\mathcal{D} as well as limits of shape 𝒟\mathcal{D}'. Then these limits “commute” with each other, in that for F:𝒟×𝒟𝒞F \;\colon\; \mathcal{D} \times {\mathcal{D}'} \to \mathcal{C} a functor (hence a diagram of shape the product category), with corresponding adjunct functors (via Example )

𝒟F 𝒟[𝒟,𝒞]AAA𝒟F 𝒟[𝒟,𝒞] {\mathcal{D}'} \overset{F_{\mathcal{D}}}{\longrightarrow} [\mathcal{D},\mathcal{C}] \phantom{AAA} {\mathcal{D}} \overset{F_{\mathcal{D}'}}{\longrightarrow} [{\mathcal{D}'}, \mathcal{C}]

we have that the canonical comparison morphism

(55)limFlim 𝒟(lim 𝒟F 𝒟)lim 𝒟(lim 𝒟F 𝒟) lim F \simeq lim_{\mathcal{D}} (lim_{\mathcal{D}'} F_{\mathcal{D}} ) \simeq lim_{\mathcal{D}'} (lim_{\mathcal{D}} F_{\mathcal{D}'} )

is an isomorphism.

Proof

Since the limit-construction is the right adjoint functor to the constant diagram-functor, this is a special case of right adjoints preserve limits (Prop. ).

See limits and colimits by example for what formula (55) says for instance for the special case 𝒞=\mathcal{C} = Set.

Remark

(general non-commutativity of limits with colimits)

In general limits do not commute with colimits. But under a number of special conditions of interest they do. Special cases and concrete examples are discussed at commutativity of limits and colimits.

\,

Proposition

(pointwise expression of left adjoints in terms of limits over comma categories)

A functor R:𝒞𝒟R \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} (Def. ) has a left adjoint L:𝒟𝒞L \;\colon\; \mathcal{D} \longrightarrow \mathcal{C} (Def. ) precisely if

  1. RR preserves all limits (Def. ) that exist in 𝒞\mathcal{C};

  2. for each object d𝒟d \in \mathcal{D}, the limit (Def. ) of the canonical functor (22) out of the comma category (Example )

    d/R𝒞 d/R \longrightarrow \mathcal{C}

    exists.

In this case the value of the left adjoint LL on dd is given by that limit:

(56)L(d)lim(c,d f R(c))d/Rc L(d) \;\simeq\; \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c
Proof

First assume that the left adjoint exist. Then

  1. RR is a right adjoint and hence preserves limits since all right adjoints preserve limits (Prop. );

  2. by Prop. the adjunction unit provides a universal morphism η d\eta_d into L(d)L(d), and hence, by Prop. , exhibits (L(d),η d)(L(d), \eta_d) as the initial object of the comma category d/Rd/R. The limit over any category with an initial object exists, as it is given by that initial object.

Conversely, assume that the two conditions are satisfied and let L(d)L(d) be given by (56). We need to show that this yields a left adjoint.

By the assumption that RR preserves all limits that exist, we have

(57)R(L(d)) =R(lim(c,d f R(c))d/Rc) lim(c,d f R(c))d/RR(c) \array{ R(L(d)) & = R\left( \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} c \right) \\ & \simeq \underset{\underset{ \left( c, \array{ d \\ \downarrow^{\mathrlap{f}} \\ R(c) } \right) \in d/R }{\longleftarrow}}{\lim} R(c) }

Since the dfR(d)d \overset{f}{\to} R(d) constitute a cone over the diagram of the R(d)R(d), there is universal morphism

dAAη dAAR(L(d)). d \overset{\phantom{AA} \eta_d \phantom{AA}}{\longrightarrow} R(L(d)) \,.

By Prop. it is now sufficient to show that η d\eta_d is a universal morphism into L(d)L(d), hence that for all c𝒞c \in \mathcal{C} and dgR(c)d \overset{g}{\longrightarrow} R(c) there is a unique morphism L(d)f˜cL(d) \overset{\widetilde f}{\longrightarrow} c such that

d η d f R(L(d)) AAR(f˜)AA R(c) L(d) AAf˜AA c \array{ && d \\ & {}^{\mathllap{ \eta_d }}\swarrow && \searrow^{\mathrlap{f}} \\ R(L(d)) && \underset{\phantom{AA}R(\widetilde f)\phantom{AA}}{\longrightarrow} && R(c) \\ L(d) &&\underset{\phantom{AA}\widetilde f\phantom{AA}}{\longrightarrow}&& c }

By Prop. , this is equivalent to (L(d),η d)(L(d), \eta_d) being the initial object in the comma category c/Rc/R, which in turn is equivalent to it being the limit of the identity functor on c/Rc/R (by Example ). But this follows directly from the limit formulas (56) and (57).

Remark

(adjoint functor theorem)

Beware the subtle point in Prop. , that the comma category c/Fc/F is in general not a small category (Def. ): It has typically “as many” objects as 𝒞\mathcal{C} has, and 𝒞\mathcal{C} is not assumed to be small (while of course it may happen to be). But typical categories, such as notably the category of sets (Example ) are generally guaranteed only to admit limits over small categories. For this reason, Prop. is rarely useful for finding an adjoint functor which is not already established to exist by other means.

But there are good sufficient conditions known, on top of the condition that RR preserves limits, which guarantee the existence of an adjoint functor, after all. This is the topic of the adjoint functor theorem (one of the rare instances of useful and non-trivial theorems in mathematics for which issues of set theoretic size play a crucial role for their statement and proof).

A very special but also very useful case of the adjoint functor theorem is the existence of adjoints of base change functors between categories of (enriched) presheaves via Kan extension. This we discuss as Prop. below. Since this is most conveniently phrased in terms of special limits/colimits called ends/coends (Def. below) we first discuss these.

\,

Ends and coends

For working with enriched categories (Def. ) , a certain shape of limits/colimits (Def. ) is particularly relevant: these are called ends and coends (Def. below). We here introduce these and then derive some of their basic properties, such as notably the expression for Kan extension in terms of (co-)ends (prop. below).

\,

Definition

((co)end)

Let 𝒞\mathcal{C} be a small 𝒱\mathcal{V}-enriched category (Def. ). Let

F:𝒞 op×𝒞𝒱 F \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}

be an enriched functor (Def. ) out of the enriched product category of 𝒞\mathcal{C} with its opposite category (Def. ). Then:

  1. The coend of FF, denoted

    c𝒞F(c,c)𝒱, \overset{c \in \mathcal{C}}{\int} F(c,c) \;\in\; \mathcal{V} \,,

    is the coequalizer in 𝒱\mathcal{V} of the two actions encoded in FF via Example :

    c,d𝒞𝒞(c,d)F(d,c)AAAAAAAAc,dρ (d,c)(c)c,dρ (c,d)(d)c𝒞F(c,c)coeqc𝒞F(c,c). \underset{c,d \in \mathcal{C}}{\coprod} \mathcal{C}(c,d) \otimes F(d,c) \underoverset {\underset{\underset{c,d}{\sqcup} \rho_{(d,c)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \rho_{(c,d)}(d) }{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c \in \mathcal{C}}{\coprod} F(c,c) \overset{coeq}{\longrightarrow} \overset{c\in \mathcal{C}}{\int} F(c,c) \,.
  2. The end of FF, denoted

    c𝒞F(c,c)𝒱, \underset{c\in \mathcal{C}}{\int} F(c,c) \;\in\; \mathcal{V} \,,

    is the equalizer in 𝒱\mathcal{V} of the adjuncts of the two actions encoded in FF via example :

    c𝒞F(c,c)equc𝒞F(c,c)AAAAAAAAc,dρ˜ (c,d)(c)c,dρ˜ d,c(d)c𝒞[𝒞(c,d),F(c,d)]. \underset{c\in \mathcal{C}}{\int} F(c,c) \overset{\;\;equ\;\;}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} F(c,c) \underoverset {\underset{\underset{c,d}{\sqcup} \tilde \rho_{(c,d)}(c) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} \tilde\rho_{d,c}(d)}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c\in \mathcal{C}}{\prod} \big[ \mathcal{C}\left(c,d\right), \; F\left(c,d\right) \big] \,.
Example

For 𝒱\mathcal{V} a cosmos, let G𝒱G \in \mathcal{V} be a group object. There is the n the one-object 𝒱\mathcal{V}-enriched category BG\mathbf{B}G as in Example .

Then a 𝒱\mathcal{V}-enriched functor

(X,ρ l):BG𝒱 (X,\rho_l) \;\colon\; \mathbf{B}G \longrightarrow \mathcal{V}

is an object XF(*)𝒱X \coloneqq F(\ast) \; \in \mathcal{V} equipped with a morphism

ρ l:GXX \rho_l \;\colon\; G \otimes X \longrightarrow X

satisfying the action property. Hence this is equivalently an action of GG on XX.

The opposite category (def. ) (BG) op(\mathbf{B}G)^{op} comes from the opposite group-object

(BG) op=B(G op). (\mathbf{B}G)^{op} = \mathbf{B}(G^{op}) \,.

(The isomorphism GG opG \simeq G^{op} induces a canonical euqivalence of enriched categories (BG) opBG(\mathbf{B}G)^{op} \simeq \mathbf{B}G.)

So an enriched functor

(Y,ρ r):(BG) op𝒱 (Y,\rho_r) \;\colon\; (\mathbf{B} G)^{op} \longrightarrow \mathcal{V}

is equivalently a right action of GG.

Therefore the coend of two such functors (def. ) coequalizes the relation

(xg,y)(x,gy) (x g,\;y) \sim (x,\; g y)

(where juxtaposition denotes left/right action) and is the quotient of the plain tensor product by the diagonal action of the group GG:

*B(G +)(Y,ρ r)(*)(X,ρ l)(*)Y GX. \overset{\ast \in \mathbf{B}(G_+)}{\int} (Y,\rho_r)(\ast) \,\otimes\, (X,\rho_l)(\ast) \;\simeq\; Y \otimes_G X \,.
Example

(enriched natural transformations as ends)

Let 𝒞\mathcal{C} be a small enriched category (Def. ). For F,G:𝒞𝒱 F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{V} two enriched presheaves (Example ), the end (def. ) of the internal-hom-functor

[F(),G()]:𝒞 op×𝒞𝒱 [F(-),G(-)] \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}

is an object of 𝒱\mathcal{V} whose underlying set (Example ) is the set of enriched natural transformations FGF \Rightarrow G (Def. )

Hom 𝒱(1,(c𝒞[F(c),G(c)]))Hom [𝒞,𝒱](F,G). Hom_{\mathcal{V}}\left(1, \left( \underset{c \in \mathcal{C}}{\int} \big[ F(c),G(c) \big] \right) \right) \;\simeq\; Hom_{[\mathcal{C}, \mathcal{V}]}(F,G) \,.
Proof

The underlying pointed set functor Hom 𝒱(1,):𝒱SetHom_{\mathcal{V}}(1,-)\colon \mathcal{V}\to Set preserves all limits, since hom-functors preserve limits (Prop. ). Therefore there is an equalizer diagram in Set of the form

Hom 𝒱(1,(c𝒞[F(c),G(c)]))equc𝒞Hom 𝒱(F(c),G(c))AAAAAAAAc,dU(ρ˜ (c,d)(c))c,dU(ρ˜ d,c(d))c,d𝒞Hom 𝒱(𝒞(c,d),[F(c),G(d)]), Hom_{\mathcal{V}}\left(1, \left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \right) \overset{equ}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} Hom_{\mathcal{V}}(F(c),G(c)) \underoverset {\underset{\underset{c,d}{\sqcup} U(\tilde \rho_{(c,d)}(c)) }{\longrightarrow}} {\overset{\underset{c,d}{\sqcup} U(\tilde\rho_{d,c}(d))}{\longrightarrow}} {\phantom{AAAAAAAA}} \underset{c,d\in \mathcal{C}}{\prod} Hom_{\mathcal{V}}( \mathcal{C}(c,d), [F(c),G(d)] ) \,,

where we used Example to identify underlying sets of internal homs with hom-sets.

Here the object in the middle is just the set of indexed sets of component morphisms {F(c)η cG(c)} c𝒞\left\{ F(c)\overset{\eta_c}{\to} G(c)\right\}_{c\in \mathcal{C}}. The two parallel maps in the equalizer diagram take such a collection to the indexed set of composites (45) and (46). Hence that these two are equalized is precisely the condition that the indexed set of components constitutes an enriched natural transformation.

Conversely, example says that ends over bifunctors of the form [F(),G())][F(-),G(-))] constitute hom-spaces between pointed topologically enriched functors:

Definition

(enriched presheaf category)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} be a small 𝒱\mathcal{V}-enriched category (Def. ).

Then the 𝒱\mathcal{V}-enriched presheaf category [𝒞,𝒱][\mathcal{C}, \mathcal{V}] is 𝒱\mathcal{V}-enriched functor category from 𝒞\mathcal{C} to 𝒱\mathcal{V}, hence is the following 𝒱\mathcal{V}-enriched category (Def. )

  1. the objects are the 𝒞\mathcal{C}-enriched functors 𝒞F𝒱\mathcal{C} \overset{F}{\to}\mathcal{V} (Def. );

  2. the hom-objects are the ends

    (58)[𝒞,𝒱](F,G) c𝒞[F(c),G(c)] [\mathcal{C}, \mathcal{V}](F,G) \;\coloneqq\; \int_{c\in \mathcal{C}} [F(c),G(c)]
  3. the composition operation on these is defined to be the one induced by the composite maps

    (c𝒞[F(c),G(c)])(c𝒞[G(c),H(c)])c𝒞[F(c),G(c)][G(c),H(c)]( F(c),G(c),H(c)) c𝒞c𝒞[F(c),H(c)], \left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \otimes \left( \underset{c \in \mathcal{C}}{\int} [G(c),H(c)] \right) \overset{}{\longrightarrow} \underset{c\in \mathcal{C}}{\prod} [F(c),G(c)] \otimes [G(c),H(c)] \overset{(\circ_{F(c),G(c),H(c)})_{c\in \mathcal{C}}}{\longrightarrow} \underset{c \in \mathcal{C}}{\prod} [F(c),H(c)] \,,

    where the first morphism is degreewise given by projection out of the limits that defined the ends. This composite evidently equalizes the two relevant adjunct actions (as in the proof of example ) and hence defines a map into the end

    (c𝒞[F(c),G(c)])(c𝒞[G(c),H(c)])c𝒞[F(c),H(c)]. \left( \underset{c\in \mathcal{C}}{\int} [F(c),G(c)] \right) \otimes \left( \underset{c \in \mathcal{C}}{\int} [G(c),H(c)] \right) \longrightarrow \underset{c\in \mathcal{C}}{\int} [F(c),H(c)] \,.

By Example , the underlying plain category (Example ) of this enriched functor category is the plain functor category of enriched functors from Example .

Proposition

(enriched Yoneda lemma)

For 𝒱\mathcal{V} a cosmos (Def. ) let 𝒞\mathcal{C} be a small enriched category (Def. ). For F:𝒞𝒱F \colon \mathcal{C} \to \mathcal{V} an enriched presheaf (Example ) and for c𝒞c\in \mathcal{C} an object, there is a natural isomorphism

[𝒞,𝒱](𝒞(c,),F)F(c) [\mathcal{C}, \mathcal{V}](\mathcal{C}(c,-),\; F) \;\simeq\; F(c)

between the hom-object of the enriched functor category (Def. ), from the functor represented by cc to FF, and the value of FF on cc.

In terms of the ends (def. ) defining these hom-objects (58), this means that

d𝒞[𝒞(c,d),F(d)]F(c). \underset{d\in \mathcal{C}}{\int} [\mathcal{C}(c,d), F(d)] \;\simeq\; F(c) \,.

In this form the statement is also known as Yoneda reduction.

Now that natural transformations are expressed in terms of ends (example ), as is the enriched Yoneda lemma (prop. ), it is natural to consider the dual statement (Example ) involving coends:

Proposition

(enriched co-Yoneda lemma)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} be a small 𝒱\mathcal{V}-enriched category (Def. ). For F:𝒞𝒱F \colon \mathcal{C}\to \mathcal{V} an enriched presheaf (Def. ) and for c𝒞c\in \mathcal{C} an object, there is a natural isomorphism

F()c𝒞𝒞(c,)F(c). F(-) \simeq \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \,.

Moreover, the morphism that hence exhibits F(c)F(c) as the coequalizer of the two morphisms in def. is componentwise the canonical action

𝒞(c,d)F(c)F(d) \mathcal{C}(c,d) \otimes F(c) \longrightarrow F(d)

which is adjunct to the component map 𝒞(d,c)[F(c),F(d)]\mathcal{C}(d,c) \to [F(c),F(d)] of the enriched functor FF.

(e.g. MMSS 00, lemma 1.6)

Proof

By the definition of coends and the universal property of colimits, enriched natural transformations of the form

(c𝒞𝒞(c,)F(c))G \left( \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \right) \longrightarrow G

are in natural bijection with systems of component morphisms

𝒞(c,d)F(c)G(d) \mathcal{C}(c,d) \otimes F(c) \longrightarrow G(d)

which satisfy some compatibility conditions in their dependence on cc and dd (natural in dd and “extranatural” in cc). By the internal hom adjunction, these are in natural bijection to systems of morphisms of the form

F(c)[𝒞(c,d),G(d)] F(c) \longrightarrow [\mathcal{C}(c,d), G(d)]

satisfying the analogous compatibility conditions. By Example these are in natural bijection with systems of morphisms

F(c)[𝒞,𝒱](𝒞(c,),G()) F(c) \longrightarrow [\mathcal{C},\mathcal{V}](\mathcal{C}(c,-), G(-))

natural in cc

By the enriched Yoneda lemma (Prop. ), these, finally, are in natural bijection with systems of morphisms

F(c)G(c) F(c) \longrightarrow G(c)

natural in cc. Moreover, all these identifications are also natural in GG. Therefore, in summary, this shows that there is a natural isomorphism

Hom [𝒞,𝒱](c𝒞𝒞(c,)F(c),())Hom [𝒞,𝒱](F,()). Hom_{[\mathcal{C},\mathcal{V}]} \left( \overset{c \in \mathcal{C}}{\int} \mathcal{C}(c,-) \otimes F(c) \;,\; (-) \right) \;\simeq\; Hom_{[\mathcal{C},\mathcal{V}]} \left( F, (-) \right) \,.

With this, the ordinary Yoneda lemma (Prop. ) in the form of the Yoneda embedding of [𝒞,𝒱][\mathcal{C},\mathcal{V}] implies the required isomorphism.

Example

(co-Yoneda lemma over Set)

Consider the co-Yoneda lemma (Prop. ) in the special case 𝒱=\mathcal{V} = Set (Example ).

In this case the coequalizer in question is the set of equivalence classes of pairs

(cc 0,x)𝒞(c,c 0)F(c), ( c \overset{}{\to} c_0,\; x ) \;\; \in \mathcal{C}(c,c_0) \otimes F(c) \,,

where two such pairs

(cfc 0,xF(c)),(dgc 0,yF(d)) ( c \overset{f}{\to} c_0,\; x \in F(c) ) \,,\;\;\;\; ( d \overset{g}{\to} c_0,\; y \in F(d) )

are regarded as equivalent if there exists

cϕd c \overset{\phi}{\to} d

such that

f=gϕ,andy=ϕ(x). f = g \circ \phi \,, \;\;\;\;\;and\;\;\;\;\; y = \phi(x) \,.

(Because then the two pairs are the two images of the pair (g,x)(g,x) under the two morphisms being coequalized.)

But now considering the case that d=c 0d = c_0 and g=id c 0g = id_{c_0}, so that f=ϕf = \phi shows that any pair

(cϕc 0,xF(c)) ( c \overset{\phi}{\to} c_0, \; x \in F(c))

is identified, in the coequalizer, with the pair

(id c 0,ϕ(x)F(c 0)), (id_{c_0},\; \phi(x) \in F(c_0)) \,,

hence with ϕ(x)F(c 0)\phi(x)\in F(c_0).

As a conceptually important corollary we obtain:

Proposition

(category of presheaves is free co-completion)

For 𝒞\mathcal{C} a small category (Def. ), its Yoneda embedding 𝒞y[𝒞 op,Set]\mathcal{C} \overset{y}{\hookrightarrow} [\mathcal{C}^{op}, Set] (Prop. ) exhibits the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] (Example ) as the free co-completion of 𝒞\mathcal{C} under forming colimits (Def. ), in that it is a universal morphism, as in Def. but “up to natural isomorphism”, into a category with all colimits (by Example ) in the following sense:

  1. for 𝒟\mathcal{D} any category with all colimits (Def. );

  2. for F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} any functor;

there is a functor F˜:[𝒞 op,Set]𝒟\widetilde F \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow \mathcal{D}, unique up to natural isomorphism such that

  1. F˜\widetilde F preserves all colimits,

  2. F˜\widetilde F extends FF through the Yoneda embedding, in that the following diagram commutes, up to natural isomorphism (Def. ):

𝒞 y F [𝒞 op,Set] F˜ 𝒟 \array{ && \mathcal{C} \\ & {}^{y}\swarrow &\swArrow& \searrow^{\mathrlap{F}} \\ \mathrlap{ \!\!\!\!\!\!\!\!\!\!\!\!\! [\mathcal{C}^{op}, Set] } && \underset{ \widetilde F }{\longrightarrow} && \mathcal{D} }

Hence when interpreting presheaves as generalized spaces, this says that “generalized spaces are precisely what is obtained from allowing arbitrary gluings of ordinary spaces”, see also Remark below.

Proof

The last condition says that F˜\widetilde F is fixed on representable presheaves by

(59)F˜(y(c))F(c). \widetilde F( y(c) ) \simeq F(c) \,.

and in fact naturally so:

(60)c 1 F˜(y(c 1)) F(c 1) f F(y(f)) F(f) c 2 F˜(y(c 2)) F(c 2) \array{ c_1&& \widetilde F( y(c_1) ) &\simeq& F(c_1) \\ {}^{\mathllap{f}}\big\downarrow && {}^{\mathllap{ F(y(f)) }}\big\downarrow && \big\downarrow^{\mathrlap{ F(f) }} \\ c_2 && \widetilde F (y(c_2)) &\simeq& F(c_2) }

But the co-Yoneda lemma (Prop. ) expresses every presheaf X[𝒞 op,Set]\mathbf{X} \in [\mathcal{C}^{op}, Set] as a colimit of representable presheaves (in the special case of enrichment over SetSet, Example )

X c𝒞y(c)X(c). \mathbf{X} \;\simeq\; \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \,.

Since F˜\tilde F is required to preserve any colimit and hence these particular colimits, (59) implies that F˜\widetilde F is fixed to act, up to isomorphism, as

F˜(X) =F˜( c𝒞y(c)X(c)) c𝒞F(c)X(c)𝒟 \begin{aligned} \widetilde F(\mathbf{X}) & = \widetilde F \left( \int^{c \in \mathcal{C}} y(c) \cdot \mathbf{X}(c) \right) & \coloneqq \int^{c \in \mathcal{C}} F(c) \cdot \mathbf{X}(c) \;\;\;\;\in \mathcal{D} \end{aligned}

(where the colimit on the right is computed in 𝒟\mathcal{D}!).

Remark

The statement of the co-Yoneda lemma in prop. is a kind of categorification of the following statement in analysis (whence the notation with the integral signs):

For XX a topological space, f:Xf \colon X \to\mathbb{R} a continuous function and δ(,x 0)\delta(-,x_0) denoting the Dirac distribution, then

xXδ(x,x 0)f(x)=f(x 0). \int_{x \in X} \delta(x,x_0) f(x) = f(x_0) \,.

It is this analogy that gives the name to the following statement:

Proposition

(Fubini theorem for (co)-ends)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞 1,𝒞 2\mathcal{C}_1, \mathcal{C}_2 be two 𝒱\mathcal{V}-enriched categories (Def. ) and

F:(𝒞 1×𝒞 2) op×(𝒞 1×𝒞 2)𝒱 F \;\colon\; \left( \mathcal{C}_1\times\mathcal{C}_2 \right)^{op} \times (\mathcal{C}_1 \times\mathcal{C}_2) \longrightarrow \mathcal{V}

a 𝒱\mathcal{V}-enriched functor (Def. ) from the product category with opposite categories (Def. ), as shown.

Then its end and coend (def. ) is equivalently formed consecutively over each variable, in either order:

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)) \overset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_1}{\int} \overset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \overset{c_2}{\int} \overset{c_1}{\int} F((c_1,c_2), (c_1,c_2))

and

(c 1,c 2)F((c 1,c 2),(c 1,c 2))c 1c 2F((c 1,c 2),(c 1,c 2))c 2c 1F((c 1,c 2),(c 1,c 2)). \underset{(c_1,c_2)}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_1}{\int} \underset{c_2}{\int} F((c_1,c_2), (c_1,c_2)) \simeq \underset{c_2}{\int} \underset{c_1}{\int} F((c_1,c_2), (c_1,c_2)) \,.
Proof

Because limits commute with limits, and colimits commute with colimits.

Remark

(internal hom preserves ends)

Let 𝒱\mathcal{V} be a cosmos (Def. ). Since the internal hom-functor in 𝒱\mathcal{V} (Def. ) preserves limits in both variables (Prop. ), in particular it preserves ends (Def. ) in the second variable, and sends coends in the second variable to ends:

For all small 𝒞\mathcal{C}-enriched categories, 𝒱\mathcal{V}-enriched functors F:𝒞 op𝒞𝒱F \;\colon\; \mathcal{C}^{op} \otimes \mathcal{C} \to \mathcal{V} (Def. ) and all objects X𝒱X \in \mathcal{V} we have natural isomorphisms

[X, c𝒞F(c,c)] c𝒞[X,F(c,c)] \left[ X , \int^{c \in \mathcal{C}} F(c,c) \right] \;\simeq\; \int^{c \in \mathcal{C}} \left[ X, F(c,c) \right]

and

[ c𝒞F(c,c),X] c𝒞[F(c,c),X]. \left[ \int_{c \in \mathcal{C}} F(c,c) , X \right] \;\simeq\; \int^{c \in \mathcal{C}} \left[ F(c,c), X \right] \,.

With this coend calculus in hand, there is an elegant proof of the defining universal property of the smash tensoring and powering enriched presheaves

Definition

(tensoring and powering of enriched presheaves)

Let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category, def. , with [𝒞,𝒱][\mathcal{C}, \mathcal{V}] its functor category of enriched functors (Example ).

  1. Define a functor

    ()():[𝒞,𝒱]×𝒱[𝒞,𝒱] (-)\cdot(-) \;\colon\; [\mathcal{C}, \mathcal{V}] \times \mathcal{V} \longrightarrow [\mathcal{C}, \mathcal{V}]

    by forming objectwise tensor products

    FX:cF(c)X. F \cdot X \;\colon\; c \mapsto F(c) \otimes X \,.

    This is called the tensoring of [𝒞,𝒱][\mathcal{C}, \mathcal{V}] over 𝒱\mathcal{V}.

  2. Define a functor

    () ():𝒱 op×[𝒞,𝒱][𝒞,𝒱] (-)^{(-)} \;\colon\; \mathcal{V}^{op} \times [\mathcal{C}, \mathcal{V}] \longrightarrow [\mathcal{C}, \mathcal{V}]

    by forming objectwise internal homs (Def. )

    F X:c[X,F(c)]. F^X \;\colon\; c \mapsto [X,F(c)] \,.

    This is called the powering of [𝒞,𝒱][\mathcal{C}, \mathcal{V}] over 𝒱\mathcal{V}.

Proposition

(universal property of tensoring and powering of enriched presheaves)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C} be a small 𝒱\mathcal{V}-enriched category (Def. ), with [𝒞,𝒱][\mathcal{C},\mathcal{V}] the corresponding enriched presheaf category.

Then there are natural isomorphisms

[𝒞,𝒱](XK,Y)[K,([𝒞,𝒱](X,Y))] [\mathcal{C}, \mathcal{V}]( X \cdot K ,\, Y ) \;\simeq\; [K,\big( [\mathcal{C}, \mathcal{V}]\left( X, Y \right) \big)]

and

[𝒞,𝒱](X,Y K)[K,([𝒞,𝒞](X,Y))] [\mathcal{C}, \mathcal{V}]\left( X,\, Y^K \right) \;\simeq\; [K, \big( [\mathcal{C}, \mathcal{C}](X,Y) \big) ]

for all X,Y[𝒞,𝒱]X,Y \in [\mathcal{C}, \mathcal{V}] and all K𝒞K \in \mathcal{C}, where () K(-)^K is the powering and ()K(-)\cdot K the tensoring from Def. .

In particular there is the composite natural isomorphism

[𝒞,𝒱](XK,Y)[𝒞,𝒱](X,Y K) [\mathcal{C}, \mathcal{V}](X \cdot K, Y) \;\simeq\; [\mathcal{C}, \mathcal{V}]\left( X, Y^K \right)

exhibiting a pair of adjoint functors

[𝒞,𝒱]() K()K[𝒞,𝒱]. [\mathcal{C}, \mathcal{V}] \underoverset {\underset{(-)^K}{\longrightarrow}} {\overset{(-) \cdot K}{\longleftarrow}} {\bot} [\mathcal{C}, \mathcal{V}] \,.
Proof

Via the end-expression for [𝒞,𝒱](,)[\mathcal{C}, \mathcal{V}](-,-) from Example , and the fact (remark ) that the internal hom-functor ends in the second variable, this reduces to the fact that [,][-,-] is the internal hom in the closed monoidal category 𝒱\mathcal{V} (Example ) and hence satisfies the internal tensor/hom-adjunction isomorphism (prop. ):

[𝒞,𝒱](XK,Y) =c[(XK)(c),Y(c)] c[X(c)K,Y(x)] c[K,[X(c),Y(c)]] [K,c[X(c),Y(c)]] =[K,([𝒞,𝒱](X,Y))] \begin{aligned} [\mathcal{C}, \mathcal{V}](X \cdot K, Y) & = \underset{c}{\int} [ (X \cdot K)(c), Y(c) ] \\ & \simeq \underset{c}{\int} [X(c) \otimes K, Y(x)] \\ & \simeq \underset{c}{\int} [K,[X(c), Y(c)]] \\ & \simeq [K, \underset{c}{\int} [X(c),Y(c)]] \\ & = [K,\left( [\mathcal{C},\mathcal{V}](X,Y)\right)] \end{aligned}

and

[𝒞,𝒱](X,Y K) =c[X(c),Y K(c)] c[X(c),[K,Y(c)]] c[X(c)K,Y(c)] c[K,[X(c),Y(c)]] [K,c[X(c),Y(c)]] [K,[𝒞,𝒱](X,Y]. \begin{aligned} [\mathcal{C}, \mathcal{V}](X, Y^K) & = \underset{c}{\int} [X(c), Y^K(c)] \\ & \simeq \underset{c}{\int} [ X(c), [K,Y(c)] ] \\ & \simeq \underset{c}{\int} [ X(c) \otimes K, Y(c) ] \\ & \simeq \underset{c}{\int} [K, [X(c),Y(c)]] \\ & \simeq [K, \underset{c}{\int} [X(c),Y(c)] ] \\ & \simeq [K, [\mathcal{C}, \mathcal{V}](X,Y] \,. \end{aligned}

\,

Tensoring and cotensoring

We make explicit the general concept of which Prpp. provides a key class of examples:

\,

Definition

(tensoring and cotensoring)

For 𝒱\mathcal{V} a cosmos (Def. ) let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ). Recall the enriched hom-functors (Example )

𝒞(,):𝒞 op×𝒞𝒱 \mathcal{C}(-,-) \;\colon\; \mathcal{C}^{op} \times \mathcal{C} \longrightarrow \mathcal{V}

and (via Example )

𝒱(,)=[,]:𝒱 op×𝒱𝒱. \mathcal{V}(-,-) = [-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{V} \longrightarrow \mathcal{V} \,.
  1. A powering (or cotensoring) of 𝒞\mathcal{C} over 𝒱\mathcal{V} is

    1. a functor (Def. )

      [,]:𝒱 op×𝒞𝒞 [-,-] \;\colon\; \mathcal{V}^{op} \times \mathcal{C} \longrightarrow \mathcal{C}
    2. for each v𝒱v \in \mathcal{V} a natural isomorphism (Def. ) of the form

      (61)𝒱(v,𝒞(c 1,c 2))𝒞(c 1,[v,c 2]) \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2])
  2. A copowering (or tensoring) of 𝒞\mathcal{C} over 𝒱\mathcal{V} is

    1. a functor (Def. )

      ()():𝒱×𝒞𝒞 (-)\otimes(-) \;\colon\; \mathcal{V} \times \mathcal{C} \longrightarrow \mathcal{C}
    2. for each v𝒱v \in \mathcal{V} a natural isomorphism (Def. ) of the form

      (62)𝒞(vc 1,c 2)𝒱(v,𝒞(c 1,c 2)) \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) )

If 𝒞\mathcal{C} is equipped with a (co-)powering it is called (co-)powered over 𝒱\mathcal{V}.

Proposition

(tensoring left adjoint to cotensoring)

For 𝒱\mathcal{V} a cosmos (Def. ) let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ).

If 𝒞\mathcal{C} is both tensored and cotensored over 𝒱\mathcal{V} (Def. ), then for fixed v𝒱v \in \mathcal{V} the operations of tensoring with vv and of cotensoring with 𝒱\mathcal{V} form a pair of adjoint functors (Def. )

𝒞AAAA[v,]v()𝒞 \mathcal{C} \underoverset {\underset{ [v,-] }{\longrightarrow}} {\overset{ v \otimes (-) }{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} \mathcal{C}
Proof

The hom-isomorphism (10) characterizing the pair of adjoint functors is provided by the composition of the natural isomorphisms (61) and (62):

𝒞(vc 1,c 2)𝒱(v,𝒞(c 1,c 2))𝒞(c 1,[v,c 2]) \array{ \mathcal{C}(v \otimes c_1, c_2) \;\simeq\; \mathcal{V}(v, \mathcal{C}(c_1,c_2) ) \;\simeq\; \mathcal{C}(c_1,[v,c_2]) }
Proposition

(in tensored and cotensored categories initial/terminal objects are enriched initial/terminal)

For 𝒱\mathcal{V} a cosmos (Def. ) let 𝒞\mathcal{C} be a 𝒱\mathcal{V}-enriched category (Def. ).

If 𝒞\mathcal{C} is both tensored and cotensored over 𝒱\mathcal{V} (Def. ) then

  1. an initial object \emptyset (Def. ) of the underlying category of 𝒞\mathcal{C} (Example ) is also enriched initial, in that the hom-object out of it is the terminal object *\ast of 𝒱\mathcal{V}

    𝒞(,c)* \mathcal{C}(\emptyset, c) \;\simeq\; \ast
  2. a terminal object *\ast (Def. ) of the underlying category of 𝒞\mathcal{C} (Example ) is also enriched terminal, in that the hom-object into it is the terminal object of 𝒱\mathcal{V}:

    𝒞(c,*)* \mathcal{C}(c, \ast) \;\simeq\; \ast
Proof

We discuss the first claim, the second is formally dual.

By prop. , tensoring is a left adjoint. Since left adjoints preserve colimits (Prop. ), and since an initial object is the colimit over the empty diagram (Example ), it follows that

v v \otimes \emptyset \;\simeq\; \emptyset

for all v𝒱v \in \mathcal{V}, in particular for 𝒱\emptyset \in \mathcal{V}. Therefore the natural isomorphism (62) implies for all v𝒱v \in \mathcal{V} that

𝒞(,c)𝒞(,c)𝒱(,𝒞(,c))* \mathcal{C}(\emptyset, c) \;\simeq\; \mathcal{C}( \emptyset \otimes \emptyset, c ) \;\simeq\; \mathcal{V}( \emptyset, \mathcal{C}(\emptyset, c) ) \;\simeq\; \ast

where in the last step we used that the internal hom 𝒱(,)=[,]\mathcal{V}(-,-) = [-,-] in 𝒱\mathcal{V} sends colimits in its first argument to limits (Prop. ) and used that a terminal object is the limit over the empty diagram (Example ).

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Kan extensions

Proposition

(Kan extension)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞,𝒟\mathcal{C}, \mathcal{D} be small 𝒱\mathcal{V}-enriched categories (Def. ) and let

p:𝒞𝒟 p \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

be a 𝒱\mathcal{V}-enriched functor (Def. ). Then precomposition with pp constitutes a functor between the corresponding 𝒱\mathcal{V}-enriched presheaf categories (Def. )

(63)p *:[𝒟,𝒱] [𝒞,𝒱] G Gp p^\ast \;\colon\; \array{ [\mathcal{D}, \mathcal{V}] &\longrightarrow& [\mathcal{C}, \mathcal{V}] \\ G &\mapsto& G \circ p }
  1. This enriched functor p *p^\ast (63) has an enriched left adjoint Lan pLan_p (Def. ), called left Kan extension along pp

    [𝒟,𝒱]p *Lan p[𝒞,𝒱] [\mathcal{D}, \mathcal{V}] \underoverset {\underset{p^\ast}{\longrightarrow}} {\overset{Lan_p }{\longleftarrow}} {\bot} [\mathcal{C}, \mathcal{V}]

    which is given objectwise by the coend (def. ):

    (64)(Lan pF):dc𝒞𝒟(p(c),d)F(c). (Lan_p F) \;\colon\; d \;\mapsto \; \overset{c\in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \otimes F(c) \,.
  2. The enriched functor p *p^\ast (63) has an enriched right adjoint Ran pRan_p (Def. ), called right Kan extension along pp

    [𝒞,𝒱]Ran pp *[𝒟,𝒱] [\mathcal{C}, \mathcal{V}] \underoverset {\underset{Ran_p}{\longrightarrow}} {\overset{p^\ast}{\longleftarrow}} {\bot} [\mathcal{D}, \mathcal{V}]

    which is given objectwise by the end (def. ):

    (65)(Ran pF):dc𝒞[𝒟(d,p(c)),F(c)]. (Ran_p F) \;\colon\; d \;\mapsto \; \underset{c\in \mathcal{C}}{\int} [\mathcal{D}(d,p(c)), F(c)] \,.

In summary, this means that the enriched functor

𝒞p𝒟 \mathcal{C} \overset{p}{\longrightarrow} \mathcal{D}

induces, via Kan extension, an adjoint triple (Remark ) of enriched functors

(66)Lan pp *Ran p:[𝒞,𝒱][𝒟,𝒱]. Lan_p \;\dashv\; p^\ast \;\dashv\; Ran_p \;\colon\; [\mathcal{C},\mathcal{V}] \leftrightarrow [\mathcal{D}, \mathcal{V}] \,.
Proof

Use the expression of enriched natural transformations in terms of coends (example and def. ), then use the respect of [,][-,-] for ends/coends (remark ), use the internal-hom adjunction (38), use the Fubini theorem (prop. ) and finally use Yoneda reduction (prop. ) to obtain a sequence of natural isomorphisms as follows:

[𝒟,𝒱](Lan pF,G) =d𝒟[(Lan pF)(d),G(d)] =d𝒟[c𝒞𝒟(p(c),d)F(c),G(d)] d𝒟c𝒞[𝒟(p(c),d)F(c),G(d)] c𝒞d𝒟[F(c),[𝒟(p(c),d),G(d)]] c𝒞[F(c),d𝒟[𝒟(p(c),d),G(d)]] c𝒞[F(c),G(p(c))] =[𝒞,𝒱](F,p *G). \begin{aligned} [\mathcal{D}, \mathcal{V}]( Lan_p F, \, G ) & = \underset{d \in \mathcal{D}}{\int} [ (Lan_p F)(d), \, G(d) ] \\ & = \underset{d\in \mathcal{D}}{\int} \left[ \overset{c \in \mathcal{C}}{\int} \mathcal{D}(p(c),d) \otimes F(c) ,\; G(d) \right] \\ &\simeq \underset{d \in \mathcal{D}}{\int} \underset{c \in \mathcal{C}}{\int} [ \mathcal{D}(p(c),d) \otimes F(c) \,,\; G(d) ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} \underset{d\in \mathcal{D}}{\int} [F(c), [ \mathcal{D}(p(c),d) , \, G(d) ] ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} [F(c), \underset{d\in \mathcal{D}}{\int} [ \mathcal{D}(p(c),d) , \, G(d) ] ] \\ & \simeq \underset{c\in \mathcal{C}}{\int} [ F(c), G(p(c)) ] \\ & = [\mathcal{C}, \mathcal{V}](F,p^\ast G) \end{aligned} \,.

and similarly:

[𝒟,𝒱](G,Ran pF) d𝒟[G(d),(Ran pF)(d),] d𝒟[G(d),c𝒞[𝒟(d,p(c)),F(c)]] d𝒟c𝒞[G(d)𝒟(d,p(c)),F(c)] c𝒞[d𝒟G(d)𝒟(d,p(c)),F(c)] c𝒟[G(p(c)),F(c)] [𝒞,𝒱](p *G,F) \begin{aligned} [\mathcal{D}, \mathcal{V}]( G,\, Ran_p F ) & \simeq \underset{d \in \mathcal{D}}{\int} [ G(d) ,\, (Ran_p F)(d), \, ] \\ & \simeq \underset{d \in \mathcal{D}}{\int} \left[ G(d) ,\, \underset{c\in \mathcal{C}}{\int} [\mathcal{D}(d,p(c)), F(c)] \right] \\ & \simeq \underset{d \in \mathcal{D}}{\int} \underset{c\in \mathcal{C}}{\int} \left[ G(d) \otimes \mathcal{D}(d,p(c)),\, F(c) \right] \\ & \simeq \underset{c\in \mathcal{C}}{\int} \left[ \overset{d \in \mathcal{D}}{\int} G(d) \otimes \mathcal{D}(d,p(c)),\, F(c) \right] \\ & \simeq \underset{c \in \mathcal{D}}{\int} \left[ G(p(c)),\, F(c) \right] \\ & \simeq [\mathcal{C}, \mathcal{V}]( p^\ast G , F ) \end{aligned}
Example

(coend formula for left Kan extension of ordinary presheaves)

Consider the cosmos to be 𝒱=\mathcal{V} = Set, via Example , so that small 𝒱\mathcal{V}-enriched categories (Def. ) are just a plain small category (Def. ) by Example , and 𝒱\mathcal{V}-enriched presheaves (Example ) are just plain presheaves (Example ).

Then for any plain functor (Def. )

𝒞 opAApAA(𝒞) op \mathcal{C}^{op} \overset{\phantom{AA} p \phantom{AA}}{\longrightarrow} (\mathcal{C}')^{op}

the general formula (64) for left Kan extension

[𝒞 op,Set]Lan p[(𝒞) op,Set] [\mathcal{C}^{op},Set] \overset{Lan_p}{\longrightarrow} [(\mathcal{C}')^{op}, Set]

is

(Lan pF)(c) cCC(c,p(c))×F(c). (Lan_p F)(c') \simeq \int^{c \in C} C'(c', p(c)) \times F(c) \,.

Using here the Yoneda lemma (Prop. ) to rewrite F(c)Hom PSh(C)(c,F)F(c) \simeq Hom_{PSh(C)}(c,F), this is

(Lan pF)(c) cCHom C(c,p(c))×Hom PSh(C)(c,F). (Lan_p F)(c') \simeq \int^{c \in C} Hom_{C'}(c', p(c)) \times Hom_{PSh(C)}(c,F) \,.

Hence this coend-set consists of equivalence classes of pairs of morphisms

(cp(c),cF) (c' \to p(c), c \to F)

where two such are regarded as equivalent whenever there is f:c 1c 2f \colon c'_1 \to c'_2 such that

c p(c 1) p(f) p(c 2) c 1 f c 2 F. \array{ && c' \\ & \swarrow && \searrow \\ p(c_1) && \stackrel{p(f)}{\longrightarrow} && p(c_2) \\ c_1 && \stackrel{f}{\longrightarrow} && c_2 \\ & \searrow && \swarrow \\ && F } \,.

This is particularly suggestive when pp is a full subcategory inclusion (Def. ). For in that case we may imagine that a representative pair (cp(c),cF)(c' \to p(c), c \to F) is a stand-in for the actual pullback of elements of FF along the would-be composite “ccFc'\to c \to F”, only that this composite need not be defined. But the above equivalence relation is precisely that under which this composite would be invariant.

Further properties

We collect here further key properties of the various universal constructions considered above.

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Proposition

(left Kan extension preserves representable functors)

For 𝒱\mathcal{V} a cosmos (Def. ), let

𝒞p𝒟 \mathcal{C} \overset{p}{\longrightarrow} \mathcal{D}

be a 𝒱\mathcal{V}-enriched functor (Def. ) between small 𝒱\mathcal{V}-enriched categories (Def. ).

Then the left Kan extension Lan pLan_p (Prop. ) takes representable enriched presheaves 𝒞(c,):𝒞𝒱\mathcal{C}(c,-) \;\colon\; \mathcal{C} \to \mathcal{V} to their image under pp:

Lan p𝒞(c,)𝒟(p(c),) Lan_p \mathcal{C}(c, -) \;\simeq\; \mathcal{D}(p(c), -)

for all cCc \in C.

Proof

By the coend formula (64) we have, naturally in d𝒟d' \in \mathcal{D}, the expression

Lan p𝒞(c,):d cC𝒟(p(c),d)𝒞(c,)(c) cC𝒟(p(c),d)𝒞(c,c) 𝒟(p(c),d), \begin{aligned} Lan_p \mathcal{C}(c,-) \;\colon\; d' \mapsto & \int^{c' \in C} \mathcal{D}(p(c'), d') \otimes \mathcal{C}(c,-)(c') \\ & \simeq \int^{c' \in C} \mathcal{D}(p(c'), d') \otimes \mathcal{C}(c,c') \\ & \simeq \mathcal{D}(p(c), d') \end{aligned} \,,

where the last step is the co-Yoneda lemma (Prop. ).

Example

(Kan extension of adjoint pair is adjoint quadruple)

For 𝒱\mathcal{V} a cosmos (Def. ), let 𝒞\mathcal{C}, 𝒟\mathcal{D} be two small 𝒱\mathcal{V}-enriched categories (Def. ) and let

𝒞pq𝒟 \mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}

be a 𝒱\mathcal{V}-enriched adjunction (Def. ). Then there are 𝒱\mathcal{V}-enriched natural isomorphisms (Def. )

(q op) *Lan p op:[𝒞 op,𝒱][𝒟 op,𝒱] (q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]
(p op) *Ran q op:[𝒟 op,𝒱][𝒞 op,𝒱] (p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]

between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other (Def. ).

By essential uniqueness of adjoint functors, this means that the two adjoint triples (Remark ) given by Kan extension (66) of qq and pp

Lan q op (q op) * Ran q op Lan p op (p op) * Ran p op \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }

merge into an adjoint quadruple (Remark )

Lan q op (q op) * (p op) * Ran p op:[𝒞 op,𝒱][𝒟 op,𝒱] \array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]
Proof

For every enriched presheaf F:𝒞 op𝒱F \;\colon\; \mathcal{C}^{op} \to \mathcal{V} we have a sequence of 𝒱\mathcal{V}-enriched natural isomorphism as follows

(Lan p opF)(d) c𝒞𝒟(d,p(c))F(c) c𝒞𝒞(q(d),c)F(c) F(q(d)) =((q op) *F)(d). \begin{aligned} (Lan_{p^{op}} F)(d) & \simeq \int^{ c \in \mathcal{C} } \mathcal{D}(d,p(c)) \otimes F(c) \\ & \simeq \int^{ c \in \mathcal{C} } \mathcal{C}(q(d),c) \otimes F(c) \\ & \simeq F(q(d)) \\ & = \left( (q^{op})^\ast F\right) (d) \,. \end{aligned}

Here the first step is the coend-formula for left Kan extension (Prop. ), the second step if the enriched adjunction-isomorphism (47) for qpq \dashv p and the third step is the co-Yoneda lemma.

This shows the first statement, which, by essential uniqueness of adjoints, implies the following statements.

Proposition

(left Kan extension along fully faithful functor is fully faithful)

For 𝒱\mathcal{V} a cosmos (Def. ), let

𝒞AApAA𝒟 \mathcal{C} \overset{\phantom{AA} p \phantom{AA}}{\hookrightarrow} \mathcal{D}

be a fully faithful 𝒱\mathcal{V}-enriched functor (Def. ) between small 𝒱\mathcal{V}-enriched categories (Def. ).

Then for all c𝒞c \in \mathcal{C}

p *(Lan pc)c p^* (Lan_p c) \simeq c

and in fact the (Lan FF *)(Lan_F \dashv F^*)-unit of an adjunction is a natural isomorphism

Idp *Lan p. Id \stackrel{\simeq}{\to} p^* \circ Lan_{p} \,.

hence, by Prop. ,

[𝒞 op,Set]AALan pAA[𝒟 op,Set] [\mathcal{C}^{op}, Set] \overset{\phantom{AA} Lan_p \phantom{AA}}{\hookrightarrow} [\mathcal{D}^{op}, Set]

is a fully faithful functor.

Proof

By the coend formula (64) we have, naturally in d𝒟d' \in \mathcal{D}, the left Kan extension of any F:𝒞𝒱F \;\colon\; \mathcal{C} \to \mathcal{V} on the image of pp is

Lan pF:p(c) cC𝒟(p(c),p(c))F(c) cC𝒞(c,c)F(c) F(c), \begin{aligned} Lan_p F \;\colon\; p(c) \mapsto & \int^{c' \in C} \mathcal{D}(p(c'), p(c)) \cdot F(c') \\ & \simeq \int^{c' \in C} \mathcal{C}(c', c) \cdot F(c') \\ & \simeq F(c) \end{aligned} \,,

where in the second step we used the assumption of fully faithfulness of pp and in the last step we used the co-Yoneda lemma (Prop. ).

Lemma

(colimit of representable is singleton)

Let 𝒞\mathcal{C} be a small category (Def. ). Then the colimit of a representable presheaf (Def. ), regarded as a functor

y(c):𝒞 opSet y(c) \;\colon\; \mathcal{C}^{op} \longrightarrow Set

is the singleton set.

(67)lim𝒟 opy(c)*. \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) \;\simeq\; \ast \,.
Proof

One way to see this is to regard the colimit as the left Kan extension (Prop. ) along the unique functor 𝒞 opp*\mathcal{C}^{op} \overset{p}{\to} \ast to the terminal category (Def. ). By the formula (64) this is

lim𝒟 opy(c) c 1𝒞*(,p(c 1))const *(c 1)×y(c)(c 1) c 1𝒞const *(c 1)×𝒞(c 1,c) const *(c) * \begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c) & \simeq \int^{c_1 \in \mathcal{C}} \underset{const_\ast(c_1)}{\underbrace{\ast(-,p(c_1))}} \times y(c)(c_1) \\ & \simeq \int^{c_1 \in \mathcal{C}} const_\ast(c_1) \times \mathcal{C}(c_1,c) \\ & \simeq const_\ast(c) \\ & \simeq \ast \end{aligned}

where we made explicit the constant functor const *:𝒞Setconst_\ast \;\colon\; \mathcal{C} \to Set, constant on the singleton set *\ast, and then applied the co-Yoneda lemma (Prop. ).

Proposition

(categories with finite products are cosifted

Let 𝒞\mathcal{C} be a small category (Def. ) which has finite products. Then 𝒞\mathcal{C} is a cosifted category, equivalently its opposite category 𝒞 op\mathcal{C}^{op} is a sifted category, equivalently colimits over 𝒞 op\mathcal{C}^{op} with values in Set are sifted colimits, equivalently colimits over 𝒞 op\mathcal{C}^{op} with values in Set commute with finite products, as follows:

For X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] to functors on the opposite category of 𝒞\mathcal{C} (hence two presheaves on 𝒞\mathcal{C}, Example ) we have a natural isomorphism (Def. )

lim𝒞 op(X×Y)(lim𝒞 opX)×(lim𝒞 opY) \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) \;\simeq\; \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right)

between the colimit of their Cartesian product and the Cartesian product of their separate colimits.

Proof

First observe that for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set] two presheaves, their Cartesian product is a colimit over presheaves represented by Cartesian products in 𝒞\mathcal{C}. Explicity, using coend-notation, we have:

(68)X×Y c 1,c 2𝒞y(c 1×c 2)×X(c 1)×Y(c 2), \mathbf{X} \times \mathbf{Y} \;\simeq\; \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \,,

where y:𝒞[𝒞 op,Set]y \;\colon\; \mathcal{C} \hookrightarrow [\mathcal{C}^{op}, Set] denotes the Yoneda embedding.

This is due to the following sequence of natural isomorphisms:

(X×Y)(c) ( c 1𝒞𝒞(c,c 1)×X(c 1))×( c 2𝒞𝒞(c,c 2)×Y(c 2)) c 1𝒞 c 2𝒞𝒞(c,c 1)×𝒞(c,c 2)𝒸(c,c 1×c 2)×(X(c 1)×X(c 2)) c 1𝒞 c 2𝒞𝒞(c,c 1×c 2)×X(c 1)×X(c 2), \begin{aligned} (\mathbf{X} \times \mathbf{Y})(c) & \simeq \left( \int^{c_1 \in \mathcal{C}} \mathcal{C}(c,c_1) \times \mathbf{X}(c_1) \right) \times \left( \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_2) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \underset{ \simeq \mathcal{c}(c, c_1 \times c_2) }{ \underbrace{ \mathcal{C}(c,c_1) \times \mathcal{C}(c,c_2) }} \times \left( \mathbf{X}(c_1) \times \mathbf{X}(c_2) \right) \\ & \simeq \int^{c_1 \in \mathcal{C}} \int^{c_2 \in \mathcal{C}} \mathcal{C}(c,c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{X}(c_2) \,, \end{aligned}

where the first step expands out both presheaves as colimits of representables separately, via the co-Yoneda lemma (Prop. ), the second step uses that the Cartesian product of presheaves is a two-variable left adjoint (by the symmetric closed monoidal structure on presheaves) and as such preserves colimits (in particular coends) in each variable separately (Prop. ), and under the brace we use the defining universal property of the Cartesian products, assumed to exist in 𝒞\mathcal{C}.

With this, we have the following sequence of natural isomorphisms:

lim𝒟 op(X×Y) lim𝒟 op c 1,c 2𝒞y(c 1×c 2)×X(c 1)×Y(c 2) c 1,c 2𝒞lim𝒟 op(y(c 1×c 2)×X(c 1)×Y(c 2)) c 1,c 2𝒞(lim𝒟 opy(c 1×c 2)*)×X(c 1)×Y(c 2) c 1,c 2𝒞(X(c 1)×Y(c 2)) ( c 1𝒞X(c 1))×( c 2𝒞Y(c 2)) (lim𝒞 opX)×(lim𝒞 opY) \begin{aligned} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( \mathbf{X} \times \mathbf{Y} \right) & \simeq \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \int^{c_1,c_2 \in \mathcal{C}} y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} \left( y(c_1 \times c_2) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \underset{ \simeq \ast }{ \underbrace{ \underset{\underset{\mathcal{D}^{op}}{\longrightarrow}}{\lim} y(c_1 \times c_2) }} \right) \times \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \\ & \simeq \int^{c_1,c_2 \in \mathcal{C}} \left( \mathbf{X}(c_1) \times \mathbf{Y}(c_2) \right) \\ & \simeq \left( \int^{c_1\in \mathcal{C}} \mathbf{X}(c_1) \right) \times \left( \int^{c_2\in \mathcal{C}} \mathbf{Y}(c_2) \right) \\ & \simeq \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{X} \right) \times \left( \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \right) \end{aligned}

Here the first step is (68), the second uses that colimits commute with colimits (Prop. ), the third uses again that the Cartesian product respects colimits in each variable separately, the fourth is by Lemma , the last step is again the respect for colimits of the Cartesian product in each variable separately.

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Basic notions of Topos theory

We have explained in Remark how presheaves on a category 𝒞\mathcal{C} may be thought of as generalized spaces probe-able by the objects of 𝒞\mathcal{C}, and that two consistency conditions on this interpretation are provided by the Yoneda lemma (Prop. ) and the resulting Yoneda embedding (Prop. ). Here we turn to a third consistency condition that one will want to impose, namely a locality or gluing condition (Remark below), to be called the sheaf condition (Def. below).

More in detail, we had seen that any category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] is the free cocompletion of the given small category 𝒞\mathcal{C} (Prop. ) and hence exhibits generalized spaces X[𝒞 op,Set]\mathbf{X} \in [\mathcal{C}^{op}, Set] as being glued or generated form the “ordinary spaces” X𝒞X \in \mathcal{C}. Further conditions to be imposed now will impose relations among these generators, such as the locality relation embodied by the sheaf-condition.

It turns out that these relations are reflected by special properties of an adjunction (Def. ) that relates generalized spaces to ordinary spaces:

generalized spaces via generators and relations:

A\phantom{A}free cocompletionA\phantom{A}
A=\phantom{A}=presheavesA\phantom{A}
A\phantom{A}loc. presentable categoryA\phantom{A}A\phantom{A}sheaf toposAAAA\phantom{AAAA}
AHAAA[𝒞 op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq} [\mathcal{C}^{op},Set]AHaccessible[𝒞 op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]AHaccessibleleft exact[𝒞 op,Set]\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot} [\mathcal{C}^{op}, Set]
A\phantom{A}Prop. A\phantom{A}A\phantom{A}Def. A\phantom{A}A\phantom{A}Prop. A\phantom{A}
A\phantom{A}simplicial presheavesA\phantom{A}A\phantom{A}combinatorial model categoryA\phantom{A}A\phantom{A}model toposA\phantom{A}
AH QuAAA[𝒞 op,sSet Qu] proj\phantom{A}\mathbf{H} \underoverset{\underset{\phantom{AAA}}{\longrightarrow}}{\overset{}{\longleftarrow}}{\simeq_{Qu}} [\mathcal{C}^{op},sSet_{Qu}]_{proj}AH Quaccessible[𝒞 op,sSet Qu] proj\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}AH Quaccessibleleft exact[𝒞 op,sSet Qu] proj\phantom{A}\mathbf{H} \underoverset{\underset{\text{accessible}}{\hookrightarrow}}{\overset{\text{left exact}}{\longleftarrow}}{\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}
A\phantom{A}Example A\phantom{A}Def. A\phantom{A}Def.

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Remark

(sheaf condition as local-to-global principle for generalized spaces)

If the objects of 𝒞\mathcal{C} are thought of as spaces of sorts, as in Remark , then there is typically a notion of locality in these spaces, reflected by a notion of what it means to cover a given space by (“smaller”) spaces (a coverage, Def. below).

But if a space X𝒞X \in \mathcal{C} is covered, say by two other spaces U 1,U 2𝒞U_1, U_2 \in \mathcal{C}, via morphisms

U 1 U 2 i 1 i 2 X \array{ U_1 && && U_2 \\ & {}_{\mathllap{i_1}}\searrow && \swarrow_{\mathrlap{i_2}} \\ && X }

then this must be reflected in the behaviour of the probes of any generalized space Y\mathbf{Y} (in the sense of Remark ) by these test spaces:

For ease of discussion, suppose that there is a sense in which these two patches above intersect in XX to form a space U 1 XU 2𝒞U_1 \cap_X U_2 \in \mathcal{C}. Then locality of probes should imply that the ways of mapping U 1U_1 and U 2U_2 into Y\mathbf{Y} such that these maps agree on the intersection U 1 XU 2U_1 \cap_X U_2, should be equivalent to the ways of mapping all of XX into Y\mathbf{Y}.

locality:{maps fromU 1andU 2toY that coincide onU 1 XU 2}{maps fromXintoY} \text{locality} \;:\; \left\{ \array{ \text{maps from}\,U_1\,\text{and}\,U_2\,\text{to}\,\mathbf{Y} \\ \text{that coincide on}\,U_1 \cap_X U_2 } \right\} \;\simeq\; \left\{ \text{maps from}\,X\,\text{into}\,\mathbf{Y} \right\}

One could call this the condition of locality of probes of generalized spaces probeable by objects of 𝒞\mathcal{C}. But the established terminology is that this is the sheaf condition (74) on presheaves over 𝒞\mathcal{C}. Those presheaves which satisfy this condition are called the sheaves (Def. below).

Remark

Warning

Most (if not all) introductions to sheaf theory insist on motivating the concept from the special case of sheaves on topological spaces (Example below). This is good motivation for what Grothendieck called “petit topos”-theory. The motivation above, instead, naturally leads to the “gros topos”-perspective, as in Example below, which is more useful for discussing the synthetic higher supergeometry of physics. In fact, this is the perspective of functorial geometry that has been highlighted since Grothendieck 65, but which has maybe remained underappreciated.

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We now first introduce the sheaf-condition (Def. ) below in its traditional form via “matching families” (Def. below). Then we show (Prop. below) how this is equivalently expressed in terms of Cech groupoids (Example below). This second formulation is convenient for understanding and handling various constructions in ordinary topos theory (for instance the definition of cohesive sites) and it makes immediate the generalization to higher topos theory.

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Descent

Here we introduce the sheaf-condition (Def. below) in its component-description via matching families (Def. below). Then we consider some of the general key properties of the resulting categories of sheaves, such as notably their “convenience”, in the technical sense of Prop. below.

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Definition

(coverage and site)

Let 𝒞\mathcal{C} be a small category (Def. ). Then a coverage on 𝒞\mathcal{C} is

such that

  • for every covering {U iι iX} iI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of XX and every morphism YfXY \overset{f}{\longrightarrow} X there exists a refining covering {V jι jY} jJ\left\{ V_j \overset{\iota_j}{\to} Y \right\}_{j \in J} of YY, meaning that for each jJj \in J there exists iIi \in I and a morphism V jι j,iU iV_j \overset{\iota_{j,i}}{\to} U_i such that

    (69)fι j=ι iι j,iAAAAAAAV j ι j,i U i ι j ι i Y f X f \circ \iota_j \;=\; \iota_i \circ \iota_{j,i} \phantom{AAAAAAA} \array{ V_j &\overset{\iota_{j,i}}{\longrightarrow}& U_i \\ {}^{\mathllap{ \iota_j }}\big\downarrow && \big\downarrow{}^{\mathrlap{ \iota_i}} \\ Y &\underset{f}{\longrightarrow}& X }

A small category 𝒞\mathcal{C} equipped with a coverage is called a site.

Example

(canonical coverage on topological spaces)

The category Top of (small) topological spaces (Example ) carries a coverage (Def. ) whose coverings are the usal open covers of topological spaces.

The condition (69) on a coverage is met, since the preimages of open subsets under a continuous function ff are again open subsets, so that the preimages of an open cover consistitute an open cover of the domain, such that the commuting diagram-condition (69) is immediage.

Similarly, for XTopX \in Top a fixed topological space, there is the site Op(X)Op(X) whose underlying category is the category of opens of XX, which is the thin category (Example ) of open subsets of XX and subset inclusions, and whose coverings are again the open covers.

Example

(differentiably good open covers of smooth manifolds)

The category SmthMfd of smooth manifold (Example ) carries a coverage (Def. ), where for XSmthMfdX \in SmthMfd any smooth manifold of dimension DD \in \mathbb{N}, its coverings are collections of smooth functions from the Cartesian space D\mathbb{R}^D to XX whose image is the inclusion of an open ball.

Hence these are the usual open covers of XX, but with the extra condition that every patch is diffeomorphic to a Cartesian space (hence to a smooth open ball).

One may further constrain this and ask that also all the non-empty finite intersections of these open balls are diffeomorphic to open balls. These are the differentiably good open covers.

To see that these coverings satisfy the condition (69): The plain pullback of an open cover along any continuous function is again an open cover, just not necessarily by patches diffeomorphic to open balls. But every open cover may be refined by one that is (see at good open cover), and this is sufficient for (69).

Example is further developed in the chapters smooth sets and on smooth homotopy types.

Definition

(matching familydescent object)

Let 𝒞\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Y[𝒞 op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over 𝒞\mathcal{C}.

Given an object X𝒞X \in \mathcal{C} and a covering {U iι iX} iI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of it (Def. ) we say that a matching family (of probes of Y\mathbf{Y}) is a tuple (ϕ iY(U i)) iI(\phi_i \in \mathbf{Y}(U_i))_{i \in I} such that for all i,jIi,j \in I and pairs of morphisms U iκ iVκ jU jU_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j satisfying

(70)ι iκ i=ι jκ jAAAAAAAA V κ i κ j U i U j ι i ι j X \iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }

we have

(71)Y(κ i)(ϕ i)=Y(κ j)(ϕ j). \mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \,.

We write

(72)Match({U i} iI,Y)iY(U i)Set Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big) \subset \underset{i}{\prod} \mathbf{Y}(U_i) \;\in\; Set

for the set of matching families for the given presheaf and covering.

This is also called the descent object of Y\mathbf{Y} for descent along the covering {U iι iX}\{U_i \overset{\iota_i}{\to}X\}.

Example

(matching families that glue)

Let 𝒞\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Y[𝒞 op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over 𝒞\mathcal{C}.

Given an object X𝒞X \in \mathcal{C} and a covering {U iι iX} iI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of it (Def. ), then every element

ϕY(X) \phi \;\in\; \mathbf{Y}(X)

induces a matching family (Def. ) by

(Y(ι i)(ϕ)) iI. \big( \mathbf{Y}(\iota_i)(\phi) \big)_{i \in I} \,.

(That this indeed satisfies the matching condition follows immediately by the functoriality of Y\mathbf{Y}.)

This construction provides a function of the form

(73)Y(X)Match({U i} iI,Y) \mathbf{Y}(X) \longrightarrow Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big)

The matching families in the image of this function are hence those tuples of probes of Y\mathbf{Y} by the patches U iU_i of XX which glue to a global probe out of XX.

Definition

(sheaves and sheaf toposes)

Let 𝒞\mathcal{C} be a small category equipped with a coverage, hence a site (Def. ) and consider a presheaf Y[𝒞 op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example ) over 𝒞\mathcal{C}.

The presheaf Y\mathbf{Y} is called a sheaf if for every object X𝒞X \in \mathcal{C} and every covering {U iι iX} iI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of XX all matching families glue uniquely, hence if the comparison morphism (73) is a bijection

(74)Y(X)Match({U i} iI,Y). \mathbf{Y}(X) \overset{\simeq}{\longrightarrow} Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big) \,.

The full subcategory (Example ) of the category of presheaves over a given site 𝒞\mathcal{C}, on those that are sheaves is the category of sheaves, denoted

(75)Sh(𝒞)AAιAA[𝒞 op,Set]. Sh(\mathcal{C}) \overset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow} [\mathcal{C}^{op}, Set] \,.

A category which is equivalent (Def. ) to a category of sheaves is called a sheaf topos, or often just topos, for short.

For H 1\mathbf{H}_1 and H 2\mathbf{H}_2 two such sheaf toposes, a homomorphism f:H 1H 2f \;\colon\; \mathbf{H}_1 \to \mathbf{H}_2 between them, called a geometric morphism is an adjoint pair of functors (Def. )

(76)H 1AAf *AAAAf *AAH 2 \mathbf{H}_1 \underoverset {\underset{ \phantom{AA} f_\ast \phantom{AA} }{\longrightarrow}} \overset{ \phantom{AA} f^\ast \phantom{AA} }{\longleftarrow} {} \mathbf{H}_2

such that

Hence there is a category Topos, whose objects are sheaf toposes and whose morphisms are geometric morphisms.

Example

(global sections geometric morphism)

Let H\mathbf{H} be a sheaf topos (Def. ). Then there is a geometric morphism (76) to the category of sets (Example ), unique up to natural isomorphism (Def. ):

HAAΓAALSet. \mathbf{H} \underoverset {\underset{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow}} {\overset{L}{\hookleftarrow}} {\bot} Set \,.

Here Γ\Gamma is called the global sections-functor.

Proof

Notice that every set SSetS \in Set is the coproduct, indexed by itself, of the terminal object *Set\ast \in Set (the singleton):

SsS*. S \;\simeq\; \underset{s \in S}{\coprod} \ast \,.

Since LL is a left adjoint, it preserves this coproduct (Prop. ). Moreover, since LL is assumed to preserve finite products, and since the terminal object is the empty product (Example ), it also preserves the terminal object. Therefore LL is fixed, up to natural isomorphism, to act as

L(S) L(sS*) sSL(*) sS*. \array{ L(S) & \simeq L \left( \underset{s \in S}{\coprod} \ast \right) \\ & \simeq \underset{s \in S}{\coprod} L(\ast) \\ & \simeq \underset{s \in S}{\coprod} \ast } \,.

This shows that LL exists and uniquely so, up to natural isomorphism. This implies the essential uniqueness of Γ\Gamma by uniqueness of adjoints (Prop. ).

Example

(trivial coverage)

For 𝒞\mathcal{C} a small category (Def. ), the trivial coverage on it is the coverage (Def. ) with no covering families at all, meaning that the sheaf condition (Def. ) over the resulting site is empty, in that every presheaf is a sheaf for this coverage.

Hence the category of presheaves [𝒞 op,Set][\mathcal{C}^{op},Set] (Example ) over a site 𝒞 triv\mathcal{C}_{triv} with trivial coverage is already the corresponding category of sheaves, hence the corresponding sheaf topos:

Sh(𝒞 triv)[𝒞 op,Set]. Sh\left( \mathcal{C}_{triv}\right) \;\simeq\; [\mathcal{C}^{op}, Set] \,.
Example

(sheaves on the terminal category are plain sets)

Consider the terminal category *\ast (Example ) equipped with its trivial coverage (Example ). Then there is a canonical equivalence of categories (Def. ) between the category of sheaves on this site (Def. ) and the category of sets (Example ):

Sh(*)Set. Sh(\ast) \;\simeq\; Set \,.

Hence the category of sets is a sheaf topos.

Example

(sheaves on a topological spacespatial petit toposes)

In the literature, the concept of (pre-)sheaf (Def. ) is sometimes not defined relative to a site, but relative to a topological space. But the latter is a special case: For XX a topological space, consider its category of open subsets Op(X)Op(X) from Example , with coverage given by the usual open covers. Then a “sheaf on this topological space” is a sheaf, in the sense of Def. , on this site of opens. One writes

Sh(X)Sh(Op(X))AA[Op(X) op,Set], Sh(X) \;\coloneqq\; Sh(Op(X)) \overset{\phantom{AA}}{\hookrightarrow} [Op(X)^{op}, Set] \,,

for short. The sheaf toposes arising this way are also called spatial toposes.

Proposition

(localic reflection)

The construction of categories of sheaves on a topological space (Example ) extends to a functor from the category Top of topological spaces and continuous functions between them (Example ) to the category Topos of sheaf toposes and geometric morphisms between them (Example ).

Sh():TopTopos. Sh(-) \;\colon\; Top \longrightarrow Topos \,.

Moreover, when restricted to sober topological spaces, this becomes a fully faithful functor, hence a full subcategory-inclusion (Def. )

Sh():SoberTopAAATopos. Sh(-) \;\colon\; SoberTop \overset{\phantom{AAA}}{\hookrightarrow} Topos \,.

More generally, this holds for locales (i.e. for “sober topological spaces not necessarily supported on points”), in which case it becomes a reflective subcategory-inclusion (Def. )

LocaleAASh()AAAAAATopos Locale \underoverset {\underset{\phantom{AA} Sh(-) \phantom{AA} }{\hookrightarrow}} {\overset{\phantom{AAAA}}{\longleftarrow}} {\bot} Topos

This says that categories of sheaves on topological spaces are but a reflection of soper topological spaces (generally: locales) and nothing more, whence they are also called petit toposes.

Example

(abelian sheaves)

In the literature, sometimes sheaves are understood by default as taking values not in the category of sets, but in the category of abelian groups. Combined with Example this means that some authors really mean “sheaf of abelian groups of the site of opens of a topological space”, when they write just “sheaf”.

But for 𝒮\mathcal{S} any mathematical structure, a sheaf of 𝒮\mathcal{S}-structured sets is equivalently an 𝒮\mathcal{S}-structure internal to the category of sheaves according to Def. . In particular sheaves of abelian groups are equivalently abelian group objects in the category of sheaves of sets as discussed here.

Example

(smooth sets)

Consider the site SmthMfd of all smooth manifolds, from Example . The category of sheaves over this (Def. ) is equivalent to the category of smooth sets, discussed in the chapter geometry of physics – smooth sets:

Sh(SmthMfd)SmoothSet. Sh(SmthMfd) \;\simeq\; SmoothSet \,.

This is a gros topos, in a sense made precise by Def. below (a cohesive topos).

Remark

(ordinary spaces and their coverings are generators and relations for generalized spaces)

Given a site 𝒞\mathcal{C} (Def. ), then its presheaf topos [𝒞 op,Set][\mathcal{C}^{op}, Set] (Example ) is the free cocompletion of the category 𝒞\mathcal{C} (Prop. ), hence the category obtained by freely forming colimits (“gluing”) of objects of 𝒞\mathcal{C}.

In contrast, the full subcategory inclusion Sh(𝒞)[𝒞 op,Set]Sh(\mathcal{C}) \hookrightarrow [\mathcal{C}^{op}, Set] enforces relations between these free colimits.

Therefore in total we may think of a sheaf topos Sh(𝒞)Sh(\mathcal{C}) as obtained by generators and relations from the objects of its site 𝒞\mathcal{C}:

  • the objects of 𝒞\mathcal{C} are the generators;

  • the coverings of 𝒞\mathcal{C} are the relations.

Proposition

(sheafification and plus construction)

Let 𝒞\mathcal{C} be a site (Def. ). Then the full subcategory-inclusion (75) of the category of sheaves over 𝒞\mathcal{C} (Def. ) into the category of presheaves (Example ) has a left adjoint (Def. ) called sheafification

Sh(𝒞)AAιAAL[𝒞 op,Set]. Sh(\mathcal{C}) \underoverset {\underset{\phantom{AA} \iota \phantom{AA}}{\hookrightarrow}} {\overset{ L }{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Set] \,.

An explicit formula for sheafification is given by applying the following “plus constructiontwice:

L(Y)(Y +) +. L(\mathbf{Y}) \simeq (\mathbf{Y}^+)^+ \,.

Here the plus construction

() +:[𝒞 op,Set][𝒞 op,Set] (-)^+ \;\colon\; [\mathcal{C}^{op}, Set] \longrightarrow [\mathcal{C}^{op}, Set]

is given by forming equivalence classes of sets of matching families (Def. ) for all possible covers (Def. )

Y +(X){{U iι iX}covering,ϕMatch({U i},Y)}/ \mathbf{Y}^+(X) \;\coloneqq\; \left\{ \{U_i \overset{\iota_i}{\to} X\} \; \text{covering} \;, \phi \in Match\left( \{U_i\}, \mathbf{Y} \right) \right\}/\sim

under the equivalence relation which identifies two such pairs if the two covers have a joint refinement such that the restriction of the two matching families to that joint refinement coincide.

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Example

(induced coverage)

Let 𝒞\mathcal{C} be a site (Def. ). Then a full subcategory (Def. )

𝒟𝒞 \mathcal{D} \hookrightarrow \mathcal{C}

becomes a site itself, whose coverage consists of those coverings {U iι iY}\{U_i \overset{\iota_i}{\to} Y\} in 𝒞\mathcal{C} that happen to be in 𝒟𝒞\mathcal{D} \hookrightarrow \mathcal{C}.

Definition

(dense subsite)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be sites (Def. ) with a a full subcategory-inclusion (Def. )

𝒟𝒞 \mathcal{D} \hookrightarrow \mathcal{C}

and regard 𝒟\mathcal{D} as equipped with the induced coverage (Def. ).

This is called a dense subsite-inclusion if every object X𝒞X \in \mathcal{C} has a covering {U iι iX} i\{U_i \overset{\iota_i}{\to} X\}_i such that for all ii the patches are in the subcategory:

U i𝒟𝒞. U_i \;\in\; \mathcal{D} \hookrightarrow \mathcal{C} \,.
Proposition

(comparison lemma)

Let 𝒟ι𝒞\mathcal{D} \overset{\iota}{\hookrightarrow} \mathcal{C} be a dense subsite inclusion (def. ). Then precomposition with ι\iota induces an equivalence of categories (Def. ) between their categories of sheaves (Def. ):

ι *:Sh(𝒞)Sh(𝒟) \iota^\ast \;\colon\; Sh(\mathcal{C}) \overset{\simeq}{\longrightarrow} Sh(\mathcal{D})
Proposition

(recognition of epi-/mono-/isomorphisms of sheaves)

Let 𝒞\mathcal{C} be a site (Def. ) with Sh(𝒞)Sh(\mathcal{C}) its category of sheaves (Def. ).

Then a morphisms f:XYf \;\colon\; \mathbf{X} \to \mathbf{Y} in Sh(𝒞)Sh(\mathcal{C}) is

  1. a monomorphism (Def. ) or isomorphism (Def. ) precisely if it is so globally in that for each object U𝒞U \in \mathcal{C} in the site, then the component f U:X(U)Y(U)f_U \colon \mathbf{X}(U) \to \mathbf{Y}(U) is an injection or bijection of sets, respectively.

  2. an epimorphism (Def. ) precisely if it is so locally, in that: for all UCU \in C there is a covering {p i:U iU} iI\{p_i : U_i \to U\}_{i \in I} such that for all iIi \in I and every element yY(U)y \in \mathbf{Y}(U) the element f(p i)(y)f(p_i)(y) is in the image of f(U i):X(U i)Y(U i)f(U_i) : \mathbf{X}(U_i) \to \mathbf{Y}(U_i).

Proposition

(epi/mono-factorization through image)

Let Sh(𝒞)Sh(\mathcal{C}) be a category of sheaves (Def. ). Then every morphism f:XYf \;\colon\; \mathbf{X} \to \mathbf{Y} factors as an epimorphism followed by a monomorphism (Def. ) uniquely up to unique isomorphism:

f:Xepiim(f)monoY. f \;\colon\; \mathbf{X} \overset{epi}{\longrightarrow} im(f) \overset{mono}{\longrightarrow} \mathbf{Y} \,.

Theobject im(f)im(f), as a subobject of Y\mathbf{Y}, is called the image of ff.

In fact this is an orthogonal factorization system, in that for every commuting square where the left morphism is an epimorphism, and the right one a monomorphism, there exists a unique lift:

(77)A AAA B epi ! mono C AAA D \array{ A &\overset{\phantom{AAA}}{\longrightarrow}& B \\ {}^{\mathllap{epi}}\big\downarrow &{}^{\exists!}\nearrow& \big\downarrow^{\mathrlap{mono}} \\ C &\underset{\phantom{AAA}}{\longrightarrow}& D }

This implies that this is a functorial factorization, in that for every commuting square

X 1 f 1 Y 1 X 2 f 2 Y 2 \array{ \mathbf{X}_1 &\overset{f_1}{\longrightarrow}& \mathbf{Y}_1 \\ \big\downarrow && \big\downarrow \\ \mathbf{X}_2 &\underset{f_2}{\longrightarrow}& \mathbf{Y}_2 }

there is an induced morphism of images such that the resulting rectangular diagram commutes:

X 1 epi im(f 1) mono Y 1 X 2 epi im(f 2) mono Y 2 \array{ \mathbf{X}_1 &\overset{epi}{\longrightarrow}& im(f_1) &\overset{mono}{\longrightarrow}& \mathbf{Y}_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ \mathbf{X}_2 &\overset{epi}{\longrightarrow}& im(f_2) &\overset{mono}{\longrightarrow}& \mathbf{Y}_2 }

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We discuss some of the key properties of sheaf toposes:

Proposition

(sheaf toposes are cosmoi)

Let 𝒞\mathcal{C} be a site (Def. ) and Sh(𝒞)Sh(\mathcal{C}) its sheaf topos (Def. ). Then:

  1. All limits exist in Sh(𝒞)Sh(\mathcal{C}) (Def. ), and they are computed as limits of presheaves, via Example :

    ι(limdX d)limdι(X d) \iota\left( \underset{\underset{d}{\longleftarrow}}{\lim} \mathbf{X}_d \right) \;\simeq\; \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d)
  2. All colimits exist in Sh(𝒞)Sh(\mathcal{C}) (Def. ) and they are given by the sheafification (Def. ) of the same colimits computed in the category of presheaves, via Example :

    limdX dL(limdι(X d)) \underset{\underset{d}{\longrightarrow}}{\lim} \mathbf{X}_d \;\simeq\; L\left( \underset{\underset{d}{\longleftarrow}}{\lim} \iota(\mathbf{X}_d) \right)
  3. The cartesian (Example ) closed monoidal category-structure (Def. ) on the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] from Example restricts to sheaves:

    Sh(𝒞)[X,]X×()Sh(𝒞) Sh(\mathcal{C}) \underoverset {\underset{[\mathbf{X}, -]}{\longrightarrow}} {\overset{\mathbf{X} \times (-)}{\longleftarrow}} {} Sh(\mathcal{C})

    In particular, for X,YSh(𝒞)\mathbf{X}, \mathbf{Y} \in Sh(\mathcal{C}) two sheaves, their internal hom [X,Y]Sh(𝒞)[\mathbf{X}, \mathbf{Y}] \in Sh(\mathcal{C}) is a sheaf given by

    [X,Y]:UHom Sh(𝒞)(y(U)X,Y), [\mathbf{X}, \mathbf{Y}] \;\colon\; U \;\mapsto\; Hom_{Sh(\mathcal{C})}( y(U) \mathbf{X}, \mathbf{Y} ) \,,

    where y(U)y(U) is the presheaf represented by U𝒞U \in \mathcal{C} (Example ).

This may be summarized by saying that every sheaf topos (in particular every category of presheaves, by Example ) is a cosmos for enriched category theory (Def. ).

Definition

(local epimorphism)

Let 𝒞\mathcal{C} be a site (Def. ). Then a morphism of presheaves over 𝒞\mathcal{C} (Example )

YAAfAAX[𝒮 op,Set] \mathbf{Y} \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{S}^{op}, Set]

is called a local epimorphism if for every object U𝒞U \in \mathcal{C}, every morphism y(U)Xy(U) \longrightarrow \mathbf{X} out of its represented presheaf (Example ) has the local lifting property through ff in that there is a covering {U iι iU}\big\{ U_i \overset{\iota_i}{\to} U \big\} (Def. ) and a commuting diagram of the form

y(U i) AAAA Y y(ι i) f y(U) AAAA X \array{ y(U_i) &\overset{\phantom{AA}\exists\phantom{AA}}{\longrightarrow}& \mathbf{Y} \\ {}^{\mathllap{y(\iota_i)}} \Big\downarrow && \Big\downarrow{}^{\mathrlap{ f }} \\ y(U) &\underset{\phantom{AAAA}}{\longrightarrow}& \mathbf{X} }

\,

Codescent

In order to understand the sheaf condition (74) better, it is useful to consider Cech groupoids (Def. below). These are really presheaves of groupoids (Def. below), a special case of the general concept of enriched presheaves. The key property of the Cech groupoid is that it co-represents the sheaf condition (Prop. below). It is in this incarnation that the concept of sheaf seamlessly generalizes to homotopy theory via “higher stacks”.

\,

Definition

(presheaves of groupoids)

For 𝒞\mathcal{C} a small category (Def. ) consider the functor category (Example ) from the opposite category of 𝒞\mathcal{C} (Example ) to the category Grpd of small groupoids (Example )

[𝒞 op,Grpd]. [\mathcal{C}^{op}, Grpd] \,.

By Example we may regard Grpd as a cosmos for enriched category theory. Since the inclusion SetGrpdSet \hookrightarrow Grpd (Example ) is a strong monoidal functor (Def. ) of cosmoi (Example ), the plain category 𝒞\mathcal{C} may be thought of as a Grpd-enriched category (Def. ) and hence a functor 𝒞 opGrpd\mathcal{C}^{op} \to Grpd is equivalently a Grpd-enriched functor (Def. ).

This means that the plain category of functors [𝒞 op,Grpd][\mathcal{C}^{op}, Grpd] enriches to Grpd-enriched category of Grpd-enriched presheaves (Example ).

Hence we may speak of presheaves of groupoids.

Remark

(presheaves of groupoids as internal groupoids in presheaves)

From every presheaf of groupoids Y[𝒞 op,Grpd]\mathbf{Y} \in [\mathcal{C}^{op}, Grpd] (Def. ), we obtain two ordinary presheaves of sets (Def. ) called the

  • presheaf of objects

    Obj Y()[𝒞 op,Set] Obj_{\mathbf{Y}(-)} \in [\mathcal{C}^{op}, Set]
  • the presheaf of morphisms

    Mor Y()x,yObj Y()Hom Y():[𝒞 op,Set] Mor_{\mathbf{Y}(-)} \;\coloneqq\; \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod} Hom_{{\mathbf{Y}(-)}} \;\colon\; [\mathcal{C}^{op}, Set]

In more abstract language this assignment constitutes an equivalence of categories

(78)[𝒞 op,Grpd] Grpd([𝒞 op,Grpd]) Y (AAAx,yObj Y()Hom Y()Mor Y() (xfy) x x xid xxxid x (xfy) y Obj Y()AAA). \array{ [\mathcal{C}^{op}, Grpd] &\overset{\simeq}{\longrightarrow}& Grpd\left( [\mathcal{C}^{op}, Grpd]\right) \\ \mathbf{Y} &\mapsto& \left( \phantom{AAA} \array{ \underset{ Mor_{\mathbf{Y}(-)} }{ \underbrace{ \underset{x,y \in Obj_{\mathbf{Y}(-)}}{\coprod} Hom_{{\mathbf{Y}(-)}} }} \\ {}^{\mathllap{ \array{(x \overset{f}{\to}y) \\ \mapsto\\ x} } }\Big\downarrow \;\;\;\; \Big\uparrow^{\mathrlap{ \array{ x \\ \mapsto \\ x \overset{id_x}{\to} x } }} \phantom{x \overset{id_x}{\to} } \Big\downarrow^{ \mathrlap{ \array{ (x \overset{f}{\to} y) \\ \mapsto \\ y } } } \\ Obj_{\mathbf{Y}(-)} } \phantom{AAA} \right) } \,.

from presheaves of groupoids to internal groupoids- in the category of presheaves over 𝒞\mathcal{C} (Def. ).

Example

(presheaves of sets form reflective subcategory of presheaves of groupoids)

Let 𝒞\mathcal{C} be a small category (Def. ). There is the reflective subcategory-inclusion (Def. ) of the category of presheaves over 𝒞\mathcal{C} (Example ) into the category of presheaves of groupoids over 𝒞\mathcal{C} (Def. )

[𝒞 op,Set]AAAAπ 0[𝒞 op,Grpd] [\mathcal{C}^{op}, Set] \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{\pi_0}{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Grpd]

which is given over each object of 𝒞\mathcal{C} by the reflective inclusion of sets into groupoids (Example ).

Example

(Cech groupoid)

Let 𝒞\mathcal{C} be a site (Def. ), and X𝒞X \in \mathcal{C} an object of that site. For each covering family {U iι iX}\{ U_i \overset{\iota_i}{\to} X\} of XX in the given coverage, the Cech groupoid is the presheaf of groupoids (Def. )

C({U i})[𝒞 op,Grpd]Grpd([𝒞 op,Set]) C(\{U_i\}) \;\in\; [\mathcal{C}^{op}, Grpd] \;\simeq\; Grpd\left( [\mathcal{C}^{op}, Set] \right)

which, regarded as an internal groupoid in the category of presheaves over 𝒞\mathcal{C}, via (78), has as presheaf of objects the coproduct

Obj C({U i})iy(U i) Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod} y(U_i)

of the presheaves represented (under the Yoneda embedding, Prop. ) by the covering objects U iU_i, and as presheaf of morphisms the coproduct over all fiber products of these:

Mor C({U i})i,jy(U i)× y(X)y(U j). Mor_{C(\{U_i\})} \;\coloneqq\; \underset{i,j}{\coprod} y(U_i) \times_{y(X)} y(U_j) \,.

This means equivalently that for any V𝒞V \in \mathcal{C} the groupoid assigned by C({U i})C(\{U_i\}) has as set of objects pairs consisting of an index ii and a morphism Vκ iU iV \overset{\kappa_i}{\to} U_i in 𝒞\mathcal{C}, and there is a unique morphism between two such objects

κ iκ j \kappa_i \longrightarrow \kappa_j

precisely if

(79)ι iκ i=ι jκ jAAAAAAAA V κ i κ j U i U j ι i ι j X \iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }

Condition (79) for morphisms in the Cech groupoid to be well-defined is verbatim the condition (70) in the definition of matching families. Indeed, Cech groupoids serve to conveniently summarize (and then generalize) the sheaf condition (Def. ):

Proposition

(Cech groupoid co-represents matching familiescodescent)

For Grpd regarded as a cosmos (Example ), and 𝒞\mathcal{C} a site (Def. ), let

Y[𝒞 op,Set][𝒞 op,Grpd] \mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]

be a presheaf on 𝒞\mathcal{C} (Example ), regarded as a Grpd-enriched presheaf via Example , let X𝒞X \in \mathcal{C} be any object and {U iι iX} i\{U_i \overset{\iota_i}{\to} X\}_i a covering family (Def. ) with induced Cech groupoid C({U i} i)C(\{U_i\}_i) (Example ).

Then there is an isomorphism

[𝒞 op,Grpd](C({U i} i),Y)Match({U i} i,Y) [\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;\simeq\; Match\left( \{U_i\}_i, \, \mathbf{Y} \right)

between the hom-groupoid of Grpd-enriched presheaves (Def. ) and the set of matching families (Def. ).

Since hence the Cech-groupoid co-represents the descent object, it is sometimes called the codescent object along the given covering.

Moreover, under this identification the canonical morphism

(80)C({U i} i)p {U i} iy(X) C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} y(X)

induces the comparison morphism (73)

[𝒞 op,Grpd](y(X),Y) Y(X) [𝒞 op,Grpd](p {U i} i,Y) [𝒞 op,Grpd](C({U i} i),Y) Match({U i} i,Y). \array{ [\mathcal{C}^{op}, Grpd]\left( y(X), \, \mathbf{Y} \right) & \simeq & \mathbf{Y}(X) \\ {}^{ \mathllap{ [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y}) } }\downarrow && \downarrow \\ [\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) &\simeq& Match\left( \{U_i\}_i, \, \mathbf{Y} \right) } \,.

In conclusion, this means that the presheaf Y\mathbf{Y} is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism:

(81)Yis a sheafAAAAAAAA[C({U i} i)p {U i}Iy(X),Y]is iso, for all covering families{U iX} \mathbf{Y} \,\text{is a sheaf} \phantom{AAAA} \Leftrightarrow \phantom{AAAA} \left[ C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}I}}{\to} y(X) \;,\; \mathbf{Y} \right] \, \text{is iso, for all covering families} \, \{U_i \to X\}

One also says in this case that Y\mathbf{Y} is a local object with respect to Cech covers/

Proof

By (58) the hom-groupoid is computed as the end

[𝒞 op,Grpd](C({U i} i),Y)= V𝒞[C({U i} i)(V),Y(V)], [\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;=\; \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] \,,

where, by Example , the “integrand” is the functor category (here: a groupoid) from the Cech groupoid at a given VV to the set (regarded as a groupoid) assigned by Y\mathbf{Y} to VV.

Since Y(V)\mathbf{Y}(V) is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the functors C({U i} i)(V)Y(V)C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V), which are equivalently those functions on sets of objects

iy(U i)(V)=Obj C({U i} i)(V)Obj Y(V)=Y(V) \underset{i}{\coprod} y(U_i)(V) = Obj_{C\left(\{U_i\}_i\right)(V)} \longrightarrow Obj_{\mathbf{Y}(V)} = \mathbf{Y}(V)

which respect the equivalence relation induced by the morphisms in the Cech groupoid at VV.

Hence the hom-groupoid is a subset of the end of these function sets:

V𝒞[C({U i} i)(V),Y(V)] V𝒞[iy(U i)(V),Y(V)] V𝒞i[y(U i)(V),Y(V)] i V𝒞[y(U i)(V),Y(V)] iY(U i) \begin{aligned} \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] & \hookrightarrow \int_{V \in \mathcal{C}} \left[ \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \int_{V \in \mathcal{C}} \underset{i}{\prod} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \int_{V \in \mathcal{C}} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \mathbf{Y}(U_i) \end{aligned}

Here we used: first that the internal hom-functor turns colimits in its first argument into limits (Prop. ), then that limits commute with limits (Prop. ), hence that in particular ends commute with products , and finally the enriched Yoneda lemma (Prop. ), which here is, via Example , just the plain Yoneda lemma (Prop. ). The end result is hence the same Cartesian product set that also the set of matching families is defined to be a subset of, in (72).

This shows that an element in V𝒞[C({U i} i)(V),Y(V)] \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] is a tuple (ϕ iY(U i)) i(\phi_i \in \mathbf{Y}(U_i))_i, subject to some condition. This condition is that for each V𝒞V \in \mathcal{C} the assignment

C({U i} i)(V) Y(V) (Vκ iU i) κ i *ϕ i=Y(κ i)(ϕ i) \array{ C\left(\{U_i\}_i\right)(V) & \longrightarrow & \mathbf{Y}(V) \\ (V \overset{\kappa_i}{\to} U_i) &\mapsto& \kappa_i^\ast \phi_i = \mathbf{Y}(\kappa_i)(\phi_i) }

constitutes a functor of groupoids.

By definition of the Cech groupoid, and since the codomain is a just set regarded as a groupoid, this is the case precisely if

Y(κ i)(ϕ i)=Y(κ j)(ϕ j)AAAAfor alli,j, \mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \phantom{AAAA} \text{for all}\, i,j \,,

which is exactly the condition (71) that makes (ϕ i) i(\phi_i)_i a matching family.

\,

Local presentation

We now discuss a more abstract characterization of sheaf toposes, in terms of properties enjoyed by the adjunction that relates them to the corresponding categories of presheaves.

Definition

(locally presentable category)

A category H\mathbf{H} (Def. ) is called locally presentable if there exists a small category 𝒞\mathcal{C} (Def. ) and a reflective subcategory-inclusion of 𝒞\mathcal{C} into its category of presheaves (Example )

HaccAALAA[𝒞 op,Set] \mathbf{H} \underoverset {\underset{\text{acc}}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} [\mathcal{C}^{op}, Set]

such that the inclusion functor is an accessible functor in that it preserves κ\kappa-filtered colimits for some regular cardinal κ\kappa.

Proposition

(sheaf toposes are equivalently the left exact reflective subcategories of presheaf toposes)

Let (𝒞,τ)(\mathcal{C}, \tau) be a site (Def. ). Then the full subcategory inclusion i:Sh(𝒞,τ)PSh(𝒞)i \colon Sh(\mathcal{C},\tau) \hookrightarrow PSh(\mathcal{C}) of its sheaf topos (Def. ) into its category of presheaves is a reflective subcategory inclusion (Def. )

Sh(𝒞,τ)ιAALAAPSh(𝒞) Sh(\mathcal{C},\tau) \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} PSh(\mathcal{C})

such that:

  1. the inclusion ι\iota is an accessible functor, thus exhibiting Sh(𝒞,τ)Sh(\mathcal{C},\tau) as a locally presentable category (Def. )

  2. the reflector L:PSh(𝒞)Sh(𝒞)L \colon PSh(\mathcal{C}) \to Sh(\mathcal{C}) (which is sheafification, Prop. ) is left exact (“lex”) in that it preserves finite limits.

Conversely, every sheaf topos arises this way. Hence sheaf toposes H\mathbf{H} are equivalently the left exact-reflectively full subcategories of presheaf toposes over some small category 𝒞\mathcal{C}:

(82)HAAaccAAAAlexAAPSh(𝒞) \mathbf{H} \underoverset {\underset{\phantom{AA}acc\phantom{AA}}{\hookrightarrow}} {\overset{\phantom{AA}lex\phantom{AA}}{\longleftarrow}} {\bot} PSh(\mathcal{C})

(e.g. Borceux 94, prop. 3.5.4, cor. 3.5.5, Johnstone, C.2.1.11)

Remark

(left exact reflections of categories of presheaves are locally presentable categories)

In the characterization of sheaf toposes as left exact reflections of categories of presheaves in Prop. , the accessibility of the inclusion, equivalently the local presentability (Def. ) is automatically implied (using the adjoint functor theorem), as indicated in (82).

\,

Gros toposes

We have seen roughly two different kinds of sheaf toposes:

Remark

(cohesive generalized spaces as foundations of geometry)

If we aim to lay foundations for geometry, then we are interested in isolating those kinds of generalized spaces which have foundational a priori meaning, independent of an otherwise pre-configured notion of space. Hence we would like to first characterize suitable gros toposes, extract concepts of space from these, and only then, possibly, consider the petit topos-reflections of these (Prop. below).

The gros toposes of such foundational generalized spaces ought to have an internal logic that knows about modalities of geometry such as discreteness or concreteness. Via the formalization of modalities in Def. this leads to the definiton of cohesive toposes (Def. , Prop. below, due to Lawvere 91, Lawvere 07).

A\phantom{A}gros toposA\phantom{A}A\phantom{A}generalized spaces obey…A\phantom{A}A\phantom{A}example:A\phantom{A}
A\phantom{A}cohesionA\phantom{A}Def. A\phantom{A}principles of differential topologyA\phantom{A}A\phantom{A}SmoothSetA\phantom{A}
A\phantom{A}elasticityDef. A\phantom{A}principles of differential geometryA\phantom{A}A\phantom{A}FormallSmoothsetA\phantom{A}
A\phantom{A}solidityA\phantom{A}Def. A\phantom{A}principles of supergeometryA\phantom{A}A\phantom{A}SuperFormalSmoothSetA\phantom{A}

\,

Cohesive toposes

Definition

(cohesive topos)

A sheaf topos H\mathbf{H} (Def. ) is called a cohesive topos if there is a quadruple (Remark ) of adjoint functors (Def. ) to the category of sets (Example )

(83)ΠDiscΓcoDisc:HAAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

such that:

  1. DiscDisc and coDisccoDisc are full and faithful functors (Def. )

  2. Π\Pi preserves finite products.

Example

(adjoint quadruple of presheaves over site with finite products)

Let 𝒞\mathcal{C} be a small category (Def. ) with finite products (hence with a terminal object *𝒞\ast \in \mathcal{C} and for any two objects X,Y𝒞X,Y \in \mathcal{C} their Cartesian product X×Y𝒞X \times Y \in \mathcal{C}).

Then there is an adjoint quadruple (Remark ) of functors between the category of presheaves over 𝒞\mathcal{C} (Example ), and the category of sets (Example )

(84)[𝒞 op,Set]AAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet [\mathcal{C}^{op}, Set] \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

such that:

  1. the functor Γ\Gamma sends a presheaf Y\mathbf{Y} to its set of global sections, which here is its value on the terminal object:

    (85)ΓY =lim𝒞Y Y(*) \begin{aligned} \Gamma \mathbf{Y} & = \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \mathbf{Y} \\ & \simeq \mathbf{Y}(\ast) \end{aligned}
  2. DiscDisc and coDisccoDisc are full and faithful functors (Def. ).

  3. Π\Pi preserves finite products:

    for X,Y[𝒞 op,Set]\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set], we have a natural bijection

    Π(X×Y)Π(X)×Π(Y). \Pi(\mathbf{X} \times \mathbf{Y}) \;\simeq\; \Pi(\mathbf{X}) \times \Pi(\mathbf{Y}) \,.

Hence the category of presheaves over a small category with finite products, hence the category of sheaves for the trivial coverage (Example ) is a cohesive topos (Def. ).

Proof

The existence of the terminal object in 𝒞\mathcal{C} means equivalently (by Example ) that there is an adjoint pair of functors between 𝒞\mathcal{C} and the terminal category (Example ):

*p𝒞 \ast \underoverset {\underset{}{\hookrightarrow}} {\overset{p}{\longleftarrow}} {\bot} \mathcal{C}

whose right adjoint takes the unique object of the terminal category to that terminal object.

From this it follows, by Example , that Kan extension produces an adjoint quadruple (Remark ) of functors between the category of presheaves [𝒞 op,Set][\mathcal{C}^{op}, Set] and [*,Set]Set[\ast, Set] \simeq Set, as shown, where

  1. Γ\Gamma is the operation of pre-composition with the terminal object inclusion *𝒞\ast \hookrightarrow \mathcal{C}

  2. DiscDisc is the left Kan extension along the inclusion *𝒞\ast \hookrightarrow \mathcal{C} of the terminal object.

The former is manifestly the operation of evaluating on the terminal object. Moreover, since the terminal object inclusion is manifestly a fully faithful functor (Def. ), it follows that also its left Kan extension DiscDisc is fully faithful (Prop. ). This implies that also coDisccoDisc is fully faithful, by (Prop. ).

Equivalently, Discp *Disc \simeq p^\ast is the constant diagram-assigning functor. By uniqueness of adjoints (Prop. ) implies that Π\Pi is the functor that sends a presheaf, regarded as a functor Y:𝒞 opSet\mathbf{Y} \;\colon\; \mathcal{C}^{op} \to Set, to its colimit

(86)Π(Y)=lim𝒞 opY. \Pi(\mathbf{Y}) \;=\; \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \mathbf{Y} \,.

The fact that this indeed preserves products follows from the assumption that 𝒞\mathcal{C} has finite products, since categories with finite products are cosifted (Prop. )

Example suggests to ask for coverages on categories with finite products which are such that the adjoint quadruple (107) on the category of presheaves (co-)restricts to the corresponding category of sheaves. The following Definition states a sufficient condition for this to be the case:

Definition

(cohesive site)

We call a site 𝒞\mathcal{C} (Def. ) cohesive if the following conditions are satisfied:

  1. The category 𝒞\mathcal{C} has finite products (as in Example ).

  2. For every covering family {U iX} i\{U_i \to X\}_i in the given coverage on 𝒞\mathcal{C} the induced Cech groupoid C({U i} i)[C op,Grpd]C(\{U_i\}_i) \in [C^{op}, Grpd] (Def. ) satisfies the following two conditions:

    1. the set of connected components of the groupoid obtained as the colimit over the Cech groupoid is the singleton:

      π 0lim𝒞 opC({U i})* \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} C(\{U_i\}) \;\simeq\; \ast
    2. the set of connected components of the groupoid obtained as the limit of the Cech groupoid is equivalent to the set of points of XX, regarded as a groupoid:

      π 0lim𝒞 opC({U i})Hom 𝒞(*,X). \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \simeq Hom_{\mathcal{C}}(\ast,X) \,.

This definition is designed to make the following true:

Proposition

(category of sheaves on a cohesive site is a cohesive topos)

Let 𝒞\mathcal{C} be a cohesive site (Def. ). Then the adjoint quadruple on the category of presheaves over 𝒞\mathcal{C}, from Example (given that a cohesive site by definition has finite products) (co-)restricts from the category of presheaves over 𝒞\mathcal{C}, to the category of sheaves (Def. ) and hence exhibits Sh(𝒞)Sh(\mathcal{C}) as a cohesive topos (Def. ):

(87)Sh(𝒞)AAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet Sh(\mathcal{C}) \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set
Proof

By example we alreaday have the analogous statement for the categories of presheaves. Hence it is sufficient to show that the functors DiscDisc and coDisccoDisc from Example factor through the definition inclusion of the category of sheaves, hence that for each set SS the presheaves Disc(S)Disc(S) and coDisc(S)coDisc(S) are indeed sheaves (Def. ).

By the formulaton of the sheaf condition via the Cech groupoid (Prop. ), and using the adjunction hom-isomorphisms (here) this is readily seen to be equivalent to the two further conditions on a cohesive site (Def. ):

Let {U iX}\{U_i \to X\} be a covering family.

The sheaf condition (81) for Disc(S)Disc(S) says that

[C({U i})p {U i} iy(X),Disc(S)] \left[ C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} y(X) \,,\, Disc(S) \right]

is an isomorphism of groupoids, which by adjunction and using (86) means equivalently that

[lim𝒞 op(C({U i}))*,S] \left[ \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast \,,\, S \right]

is an isomorphism of groupoids, where we used that colimits of representables are singletons (Lemma ) to replace lim𝒞 opy(X)*\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} y(X) \simeq \ast.

But now in this internal hom of groupoids, the set SS is really a groupoid in the image of the reflective embedding of sets into groupoids, whose left adjoint is the connected components-functor π 0\pi_0 (Example ). Hence by another adjunction isomoprhism this is equivalent to

[π 0lim𝒞 op(C({U i}))*,S] \left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast \,,\, S \right]

being an isomorphism (a bijection of sets, now). This is true for all SSetS \in Set precisely if (by the Yoneda lemma, if you wish) the morphism

π 0lim𝒞 op(C({U i}))* \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim} \left( C(\{U_i\}) \right) \to \ast

is already an isomorphism (here: bijection) itself.

Similarly, the sheaf condition (81) for coDisc(S)coDisc(S) says that

[C({U i})p {U i} iy(X),coDisc(S)] \left[ C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} y(X) \,,\, coDisc(S) \right]

is an isomorphism, and hence by adjunction and using (85), this is equivalent to

[π 0lim𝒞 opC({U i})p {U i} iHom 𝒞(*,X),S] \left[ \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X) \,,\, S \right]

being an isomorphism. This holds for all SSetS \in Set if (by the Yoneda lemma, if you wish)

π 0lim𝒞 opC({U i})p {U i} iHom 𝒞(*,X) \pi_0 \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{\lim} C(\{U_i\}) \overset{p_{\{U_i\}_i}}{\to} Hom_{\mathcal{C}}(\ast, X)

is an isomorphism.

Definition

(adjoint triple of adjoint modal operators on cohesive topos)

Given a cohesive topos (Def. ), its adjoint quadruple (Remark ) of functors to and from Set

(88)ΠDiscΓcoDisc:HAAAΠAAA AADiscAA AAAΓAAA AAcoDiscAASet \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Set

induce, by composition of functors, an adjoint triple (Remark ) of adjoint modalities (Def. ):

ʃ:HʃDiscΠ DiscΓ coDiscΓH. &#643; \dashv \flat \dashv \sharp \;\;\colon\;\; \mathbf{H} \array{ \overset{ &#643; \;\coloneqq\; Disc \circ \Pi }{\hookleftarrow} \\ \overset{\flat \;\coloneqq\; Disc \circ \Gamma }{\longrightarrow} \\ \overset{ \sharp \;\coloneqq\; coDisc\circ \Gamma }{\hookleftarrow} } \mathbf{H} \,.

Since DiscDisc and coDisccoDisc are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ&#643; \;\coloneqq\; Disc \circ \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

Proposition

(pieces have points \simeq discrete objects are concrete \simeq Aufhebung of bottom adjoint modality)

Let H\mathbf{H} be a cohesive topos (Def. ). Then the following conditions are equivalent:

  1. pieces have points: For every object XHX \in \mathbf{H}, comparison of extremes-transformation (27) for the (ʃ,)(&#643;, \dashv \flat)-adjoint modality (27), hence the \flat-counit of an adjunction composed with the ʃ-unit

    XAAϵ X AAXAAϵ X ʃAAʃX \flat X \overset{ \phantom{AA} \epsilon^\flat_X \phantom{AA} }{\longrightarrow} X \overset{ \phantom{AA} \epsilon^&#643;_X \phantom{AA} }{\longrightarrow} &#643; X

    is an epimorphism (Def. )

  2. discrete objects are concrete: For every object XHX \in \mathbf{H}, we have that its discrete object X\flat X is a concrete object (Def. ).

  3. Aufhebung of bottom adjoint modality

    The adjoint modality \flat \dashv \sharp exhibits Aufhebung (Def. ) of the bottom adjoint modality (Example ), i.e. the initial object (Def. ) is codiscrete (Def. ):

    . \sharp \emptyset \;\simeq\; \emptyset \,.
Proof

The comparison morphism ptp Hptp_{\mathbf{H}} is a special case of that discussed in Prop. . First observe, in the notation there, that

ptp His epiAAAiffAAAptp Bis epi. ptp_{\mathbf{H}} \;\; \text{is epi} \phantom{AAA} \text{iff} \phantom{AAA} ptp_{\mathbf{B}} \;\; \text{is epi} \,.

In one direction, assume that ptp Bptp_{\mathbf{B}} is an epimorphism. By (31) we have ptp H=Disc(ptp B)ptp_{\mathbf{H}} = Disc(ptp_{\mathbf{B}}), but DiscDisc is a left adjoint and left adjoints preserve monomorphisms (Prop. ).

In the other direction, assume that ptp Hptp_{\mathbf{H}} is an epimorphism. By (29) and (32) we see that ptp Bptp_{\mathbf{B}} is re-obtained from this by applying Γ\Gamma and then composition with isomorphisms. But Γ\Gamma is again a left adjoint, and hence preserves epimorphism by Prop. , as does composition with isomorphisms.

By applying (29) again, we find in particular that pieces have points is also equivalent to Πϵ DiscS \Pi \epsilon^\flat_{Disc S} being an epimorphism, for all SBS \in \mathbf{B}. But this is equivalent to

Hom B(Πϵ X ,S)=Hom H(ϵ X ,Disc(S)) Hom_{\mathbf{B}}(\Pi \epsilon^\flat_{\mathbf{X}}, S) = Hom_{\mathbf{\mathbf{H}}}(\epsilon^\flat_{\mathbf{X}}, Disc(S))

being a monomorphism for all SS (by adjunction isomorphism (10) and definition of epimorphism, Def. ).

Now by Lemma , this is equivalent to

Hom H(X,η Disc(S) ) Hom_{\mathbf{H}}( \mathbf{X}, \eta^\sharp_{Disc(S)} )

being an injection for all X\mathbf{X}, which, by Def. , is equivalent to η Disc(S) \eta^\sharp_{Disc(S)} being a monomorphism, hence to discrete objects are concrete.

This establishes the equivalence between the first two items.

Proposition

(cohesive site such that pieces have points/discrete objects are concrete)

Let 𝒞\mathcal{C} be a cohesive site (Def. ), such that

Then the cohesive topos Sh(𝒞)Sh(\mathcal{C}), according to Prop. , satisfies the equivalent conditions from Prop. :

  1. pieces have points,

  2. discrete objects are concrete.

Proof

By Prop. it is sufficient to show the second condition, hence to check that for each set SSetS \in Set, the canonical morphism

Disc(S)coDisc(S) Disc(S) \longrightarrow coDisc(S)

is a monomorphism. By Prop. this means equivalently that for each object X𝒞X \in \mathcal{C} in the site, the component function

Disc(S)(X)coDisc(S)(X) Disc(S)(X) \longrightarrow coDisc(S)(X)

is an injective function.

Now, by the proof of Prop. , this is the diagonal function

S Hom Set(Hom 𝒞(*,X),S) s const s \array{ S & \longrightarrow& Hom_{Set}\left( Hom_{\mathcal{C}}(\ast, X), S \right) \\ s &\mapsto& const_s }

This function is injective precisely if Hom 𝒞(*,X)Hom_{\mathcal{C}}(\ast, X) is non-empty, which is true by assumption.

Proposition

(quasitopos of concrete objects in a cohesive topos)

For H\mathbf{H} a cohesive topos (Def. ), write

H concAAAAH \mathbf{H}_{conc} \overset{ \phantom{AAAA} }{\hookrightarrow} \mathbf{H}

for its full subcategory (Example ) of concrete objects (Def. ).

Then there is a sequence of reflective subcategory-inclusions (Def. ) that factor the (ΓcoDisc)(\Gamma \dashv coDisc)-adjunction as

ΓcoDisc:HAAconcAA AAι concAAH concAAΓAA AAcoDiscAASet \Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set

If in addition discrete objects are concrete (Prop. ), then the full adjoint quadruple factors through the concrete objects:

a A ΓcoDisc:HAAΠAA AADiscAA AAconcAA AAι concAAH concAAΠAA AADiscAA AAΓAA AAcoDiscAASet \array{ \\ \phantom{a} \\ \phantom{A} \\ \Gamma \;\dashv\; coDisc } \;\;\colon\;\; \mathbf{H} \array{ \phantom{\overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow}} \\ \phantom{\overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow}} \\ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{ \phantom{AA} \Pi \phantom{AA} }{\longrightarrow} \\ \overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set
Proof

For the adjunction on the right, we just need to observe that for every set SSetS \in Set, the codiscrete object coDisc(S)coDisc(S) is concrete, which is immediate by idempotency of \sharp (Prop. ) and the fact that every isomorphism is also a monomorphism. Similarly, the assumption that discrete objects are concrete says exactly that also DiscDisc factors through H conc\mathbf{H}_{conc}.

For the adjunction on the left we claim that the left adjoint concconc, (to be called concretification), is given by sending each object to the image (Def. ) of its (ΓcoDisc)(\Gamma \dashv coDisc) adjunction unit η \eta^\sharp:

conc:Xim(η X ), conc \;\colon\; X \mapsto im(\eta^\sharp_X) \,,

hence to the object which exhibits the epi/mono-factorization (Prop. ) of η X \eta^\sharp_X

(89)η X :Xepiη X concconcXmonoX. \eta^\sharp_X \;\colon\; X \underoverset{epi}{ \eta^{conc}_X }{\longrightarrow} conc X \underoverset{mono}{}{\longrightarrow} \sharp X \,.

First we need to show that concXconc X, thus defined, is indeed concrete, hence that η im(η X ) \eta^\sharp_{im(\eta^\sharp_X)} is a monomorphism (Def. ). For this, consider the following naturality square (11) of the ΓcoDisc\Gamma \dashv coDisc-adjunction hom-isomorphism

(90)Hom Set(Γim(η X ),Γim(η X )) Hom H(im(η X ),im(η X )) ()Γ(η X conc) ()η X conc Hom Set(ΓX,Γim(η X )) Hom H(X,im(η X ))AAAA{id Γim(η X )} (η X conc)η X {η im(η X ) } (η X conc)η X {Γ(η X conc)} {(η X conc)isoη X =η im(η X ) η X conc} \array{ Hom_{Set}( \Gamma im(\eta^\sharp_X), \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( im(\eta^\sharp_X), \sharp im(\eta^\sharp_X) ) \\ {}^{ \mathllap{ (-) \circ \Gamma(\eta^{conc}_X) } }\big\downarrow && \big\downarrow^{ \mathrlap{ (-) \circ \eta^{conc}_X } } \\ Hom_{Set}( \Gamma X, \Gamma im(\eta^\sharp_X) ) &\simeq& Hom_{\mathbf{H}}( X, \sharp im(\eta^\sharp_X) ) } \phantom{AAAA} \array{ \left\{ id_{\Gamma im(\eta^\sharp_X)} \right\} &\longrightarrow& \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \left\{ \eta^{\sharp}_{im(\eta^\sharp_X)} \right\} \\ \big\downarrow && \phantom{\sharp(\eta^{conc}_X) \circ \eta^\sharp_{ X }} \big\downarrow \\ \left\{ \Gamma(\eta^{conc}_X) \right\} &\longrightarrow& \left\{ \underset{ iso }{ \underbrace{ \sharp(\eta^{conc}_X) }} \circ \eta^\sharp_{ X } \;=\; \eta^{\sharp}_{ im(\eta^\sharp_X) } \circ \eta^{conc}_X \right\} }

By chasing the identity morphism on Γim(η X )\Gamma im(\eta^\sharp_X) through this diagram, as shown by the diagram on the right, we obtain the equality displayed in the bottom right entry, where we used the general formula for adjuncts (Prop. ) and the definition coDiscΓ\sharp \coloneqq coDisc \circ \Gamma (Def. ).

But observe that Γ(η X conc)\Gamma (\eta^{conc}_X), and hence also (η X conc)\sharp(\eta^{conc}_X), is an isomorphism (Def. ), as indicated above: Since Γ\Gamma is both a left adjoint as well as a right adjoint, it preserves both epimorphisms as well as monomorphisms (Prop. ), hence it preserves image factorizations (Prop. ). This implies that Γη X conc\Gamma \eta^{conc}_X is the epimorphism onto the image of Γ(η X )\Gamma( \eta^\sharp_X ). But by idempotency of \sharp, the latter is an isomorphism, and hence so is the epimorphism in its image factorization.

Therefore the equality in (90) says that

η X =(isoη im(η X ) )η X conc =monoη X conc, \begin{aligned} \eta^\sharp_{ X } & = \left( iso \circ \eta^{\sharp}_{ im(\eta^\sharp_X)} \right) \circ \eta^{conc}_X \\ & = mono \circ \eta^{conc}_X \,, \end{aligned}

where in the second line we remembered that η X conc\eta^{conc}_X is, by definition, the epimorphism in the epi/mono-factorization of η X \eta^\sharp_X.

Now the defining property of epimorphisms (Def. ) allows to cancel this commmon factor on both sides, which yields

η im(η X ) =isomono=mono. \eta^{\sharp}_{ im(\eta^\sharp_X) } \;=\; iso \circ mono \;=\; mono.

This shows that concXim(η X )conc X \coloneqq im(\eta^\sharp_X) is indeed concret.

\,

It remains to show that this construction is left adjoint to the inclusion. We claim that the adjunction unit (Def. ) of (concι conc)(conc \dashv \iota_{conc}) is provided by η conc\eta^{conc} (89).

To see this, first notice that, since the epi/mono-factorization (Prop. ) is orthogonal and hence functorial, we have commuting diagrams of the form

(91)X 1 epiη X 1 conc im(η X 1 ) mono X 1 X 2 epiη X 2 conc im(η X 2 ) mono X 2 \array{ X_1 &\underoverset{epi}{\eta^{conc}_{X_1}}{\longrightarrow}& im(\eta^\sharp_{X_1}) &\underset{mono}{\longrightarrow}& \sharp X_1 \\ \big\downarrow && \big\downarrow && \big\downarrow \\ X_2 &\underoverset{epi}{\eta^{conc}_{X_2}}{\longrightarrow}& im(\eta^\sharp_{X_2}) &\underset{mono}{\longrightarrow}& \sharp X_2 }

Now to demonstrate the adjunction it is sufficient, by Prop. , to show that η conc\eta^{conc} is a universal morphism in the sense of Def. . Hence consider any morphism f:X 1X 2f \;\colon\; X_1 \to X_2 with X 2H concHX_2 \in \mathbf{H}_{conc} \hookrightarrow \mathbf{H}. Then we need to show that there is a unique diagonal morphism as below, that makes the following top left triangle commute:

X 1 AAfAA X 2 epi η X 1 conc ! mono im(η X 1 ) X 2 \array{ X_1 &\overset{\phantom{AA} f \phantom{AA}}{\longrightarrow}& X_2 \\ {}^{\mathllap{epi}}\big\downarrow^{\mathrlap{\eta^{conc}_{X_1}}} &{}^{\mathllap{\exists !}}\nearrow& \big\downarrow^{\mathrlap{mono}} \\ im(\eta^\sharp_{X_1}) &\underset{}{\longrightarrow}& \sharp X_2 }

Now, from (91), we have a commuting square as shown. Here the left morphism is an epimorphism by construction, while the right morphism is a monomorphism by assumption on X 2X_2. With this, the epi/mono-factorization in Prop. says that there is a diagonal lift which makes both triangles commute.

It remains to see that the lift is unique with just the property of making the top left triangle commute. But this is equivalently the statement that the left morphism is an epimorphism, by Def. .

The equivalence of the first two follows with (Johnstone, lemma 2.1, corollary 2.2). The equivalence of the first and the last is due to Lawvere-Menni 15, lemma 4.1, lemma 4.2.

\,

Elastic toposes

Definition

(elastic topos)

Let H red\mathbf{H}_{red} be a cohesive topos (Def. ). Then an elastic topos or differentially cohesive topos over H red\mathbf{H}_{red} is a sheaf topos H\mathbf{H} which is

  1. a cohesive topos over Set,

  2. equipped with a quadruple of adjoint functors (Def. ) to H red\mathbf{H}_{red} of the form

    H redAAι infAA AAΠ infAA AADisc infAA AAΓ infAAH \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} } \mathbf{H}
Lemma

(progression of (co-)reflective subcategories of elastic topos)

Let H\mathbf{H} be an elastic topos (Def. ) over a cohesive topos H red\mathbf{H}_{red} (Def. ):

SetAΠ redA ADisc redA AΓ redA AcoDisc redAH redAAι infAA AAΠ infAA AADisc infAA AAΓ infAA A aH Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{a} } \mathbf{H}

and write

SetAAΠAA AADiscAA AAΓAA AAcoDiscAAH Set \array{ \overset{\phantom{AA} \Pi \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}

for the adjoint quadruple exhibiting the cohesion of H\mathbf{H} itself. Then these adjoint functors arrange and decompose as in the following diagram

Proof

The identification

(DiscΓ)(Disc infDisc redΓ redΓ inf) (Disc \dashv \Gamma) \;\simeq\; ( Disc_{inf} \circ Disc_{red} \,\dashv\, \Gamma_{red} \circ \Gamma_{inf} )

follows from the essential uniqueness of the global section-geometric morphism (Example ). This implies the identifications ΠΠ redΠ inf\Pi \simeq \Pi_{red} \circ \Pi_{inf} by essential uniqueness of adjoints (Prop. ).

Definition

(adjoint modalities on elastic topos)

Given an elastic topos (differentially cohesive topos) H\mathbf{H} over H red\mathbf{H}_{red} (Def. ), composition of the functors in Lemma yields, via Prop. , the following adjoint modalities (Def. )

&:Hι infΠ inf Disc infΠ inf &Disc infΓ infH. \Re \dashv \Im \dashv \& \;\;\colon\;\; \mathbf{H} \array{ \overset{ \Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} }{\longleftarrow} \\ \overset{\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} }{\longrightarrow} \\ \overset{ \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} }{\longleftarrow} } \mathbf{H} \,.

Since ι inf\iota_{inf} and Disc infDisc_{inf} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

Proposition

(progression of adjoint modalities on elastic topos)

Let H\mathbf{H} be an elastic topos (Def. ) and consider the corresponding adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ&#643; \;\coloneqq\; Disc \circ \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def.

& ʃ * \array{ \Re &\dashv& \Im &\dashv& \& \\ && \vee && \vee \\ && &#643; &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }

where we display also the bottom adjoint modality *\emptyset \dashv \ast (Example ), for completeness.

Proof

We need to show, for all XHX \in \mathbf{H}, that

  1. X\flat X is an &\&-modal object (Def. ), hence that

    &XX \& \flat X \;\simeq\; X
  2. ʃX&#643; X is an \Im-modal object (Def. ), hence that

    ʃXX \Im &#643; X \;\simeq\; X

After unwinding the definitions of the modal operators Def. and Def. , and using their re-identification from Lemma , this comes down to the fact that

Π infDisc infidAAAandAAAΓ infDisc infid, \Pi_{inf} Disc_{inf} \;\simeq\; id \phantom{AAA} \text{and} \phantom{AAA} \Gamma_{inf} Disc_{inf} \;\simeq\; id \,,

which holds by Prop. , since Disc infDisc_{inf} is a fully faithful functor and Π inf\Pi_{inf}, Gamma infGamma_{inf} are (co-)reflectors for it, respectively:

&Disc infΓ infDiscΓ =Disc infΓ infDiscDisc infDisc redΓ =Disc infΓ infDisc infidDisc redDiscΓ Disc infDisc redDiscΓX =DiscΓ = \begin{aligned} \underset{Disc_{inf} \Gamma_{inf}}{\underbrace{\;\;\;\&\;\;\;}} \underset{Disc \Gamma }{\underbrace{\;\;\;\flat\;\;\;}} & = Disc_{inf} \Gamma_{inf} \underset{ Disc_{inf} Disc_{red} }{\underbrace{\;\;\;\Disc\;\;\;}} \Gamma \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{\simeq id}{\underbrace{\Gamma_{inf} Disc_{inf}}} Disc_{red} }} \; \Gamma \\ & \simeq \underset{ Disc }{\underbrace{ Disc_{inf} Disc_{red} }} \Gamma \mathbf{X} \\ & = Disc \Gamma \\ & = \flat \end{aligned}

and

Disc infΠ infʃDiscΠ =Disc infΠ infDiscDisc infDisc redΠ =Disc infΠ infDisc infidDisc redDiscΠ DiscΠ =ʃ \array{ \underset{ Disc_{inf} \Pi_{inf} }{ \underbrace{ \;\;\;\Im\;\;\; }} \underset{ Disc \Pi }{ \underbrace{ \;\;\;&#643;\;\;\; } } & = Disc_{inf} \Pi_{inf} \underset{ Disc_{inf} Disc_{red} }{ \underbrace{ \;\;\;Disc\;\;\; } } \Pi \\ & = \underset{ \simeq Disc }{ \underbrace{ Disc_{inf} \underset{ \simeq id }{ \underbrace{ \Pi_{inf} Disc_{inf} } } Disc_{red} } } \Pi \\ & \simeq Disc \Pi \\ & = &#643; }

\,

Solid toposes

Definition

(solid topos)

Let H bos\mathbf{H}_{bos} be an elastic topos (Def. ) over a cohesive topos H red\mathbf{H}_{red} (Def. ). Then a solid topos or super-differentially cohesive topos over H bos\mathbf{H}_{bos} is a sheaf topos H\mathbf{H}, which is

  1. a cohesive topos over Set (Def. ),

  2. an elastic topos over H red\mathbf{H}_{red},

  3. equipped with a quadruple of adjoint functors (Def. ) to H bos\mathbf{H}_{bos} of the form

    H bosAevenA AAι supAA AAΠ supAA AADisc supAAH \mathbf{H}_{bos} \array{ \overset{\phantom{A} even \phantom{A} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} } \mathbf{H}

    hence with ι sup\iota_{sup} and Disc supDisc_{sup} being fully faithful functors (Def. ).

Lemma

(progression of (co-)reflective subcategories of solid topos)

Let H\mathbf{H} be a solid topos (Def. ) over an elastic topos H red\mathbf{H}_{red} (Def. ):

SetAΠ redA ADisc redA AΓ redA AcoDisc redAH redAAι infAA AAΠ infAA AADisc infAA AAΓ infAA A AAH bosAAevenAA AAι supAA AAΠ supAA AADisc supAA AAΓ supAA A AA AAH Set \array{ \overset{\phantom{A} \Pi_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} Disc_{red} \phantom{A} }{\hookrightarrow} \\ \overset{\phantom{A} \Gamma_{red} \phantom{A} }{\longleftarrow} \\ \overset{\phantom{A} coDisc_{red} \phantom{A} }{\hookrightarrow} } \mathbf{H}_{red} \array{ \overset{\phantom{AA} \iota_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{inf} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{inf} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{inf} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} } \mathbf{H}_{bos} \array{ \overset{\phantom{AA} even \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} \iota_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Pi_{sup} \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AA} Disc_{sup} \phantom{AA} }{\hookrightarrow} \\ \overset{\phantom{AA} \Gamma_{sup} \phantom{AA} }{\longleftarrow} \\ \phantom{A} \\ \phantom{A \atop A} \\ \phantom{A \atop A} } \mathbf{H}

Then these adjoint functors arrange and decompose as shown in the following diagram:

Here the composite adjoint quadruple

SetΠΠ redΠ infΠ sup Disc=Disc supDisc infDisc red Γ=Γ supΓ infΓ red AAcoDiscAAH Set \array{ \overset{\Pi \simeq \Pi_{red}\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc = Disc_{sup} Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{\Gamma = \Gamma_{sup} \Gamma_{inf} \Gamma_{red} }{\longleftarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookrightarrow} } \mathbf{H}

exhibits the cohesion of H\mathbf{H} over Set, and the composite adjoint quadruple

H redι supι inf Π infΠ sup Disc infDisc red Γ supH \mathbf{H}_{red} \array{ \overset{\iota_{sup} \iota_{inf}}{\hookrightarrow} \\ \overset{\Pi_{inf} \Pi_{sup} }{\longleftarrow} \\ \overset{Disc_{inf} Disc_{red}}{\hookrightarrow} \\ \overset{ \Gamma_{sup} }{\longleftarrow} } \mathbf{H}

exhibits the elasticity of H\mathbf{H} over H red\mathbf{H}_{red}.

Proof

As in the proof of Prop. , this is immediate by the essential uniqueness of adjoints (Prop. ) and of the global section-geometric morphism (Example ).

Definition

(adjoint modalities on solid topos)

Given a solid topos H\mathbf{H} over H bos\mathbf{H}_{bos} (Def. ), composition of the functors in Lemma yields, via Prop. , the following adjoint modalities (Def. )

Rh:Hι supeven ι supΠ sup RhDisc supΠ supH. \rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.

Since ι sup\iota_{sup} and Disc supDisc_{sup} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ).

We pronounce these as follows:

A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

  • a \rightsquigarrow-comodal object

    Xϵ X X \overset{\rightsquigarrow}{X} \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow} X

    is called a bosonic object;

  • a RhRh-modal object

    Xη X RhRhX X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X

    is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let H\mathbf{H} be a solid topos (Def. ) and consider the adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. ,

  3. for being a solid topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ&#643; \;\coloneqq\; Disc \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι supι infΠ infΠ sup\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup} A\phantom{A}A\phantom{A} Disc supDisc infΠ infΠ sup\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup} A\phantom{A}A\phantom{A} &Disc supDisc infΓ infΓ sup \& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup} A\phantom{A}
A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def. :

id id Rh & ʃ * \array{ id &\dashv& id \\ \vee && \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee && \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && &#643; &\dashv& \flat &\dashv& \sharp \\ && && && \vee && \vee \\ && && && \emptyset &\dashv& \ast }

where we are displaying, for completeness, also the adjoint modalities at the bottom *\emptyset \dashv \ast and the top ididid \dashv id (Example ).

Proof

By Prop. , it just remains to show that for all objects XHX \in \mathbf{H}

  1. X\Im X is an RhRh-modal object, hence RhXXRh \Im X \simeq X,

  2. X\Re X is a bosonic object, hence XX\overset{\rightsquigarrow}{\Re X} \simeq \Re X.

The proof is directly analogous to that of Prop. , now using the decompositions from Lemma :

Rh =Disc supΠ supDisc supidDisc infΠ infΠ sup Disc supDisc infΠ infΠ sup = \begin{aligned} Rh \Im & = Disc_{sup} \underset{ \simeq id }{ \underbrace{ \Pi_{sup} \;\; Disc_{sup} } } Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & \simeq Disc_{sup} Disc_{inf} \Pi_{inf} \Pi_{sup} \\ & = \Im \end{aligned}

and

=ι supΠ supι supidι infΠ infΠ sub ι supι infΠ infΠ sub \begin{aligned} \rightsquigarrow \Re & = \iota_{sup} \underset{\simeq id}{\underbrace{ \Pi_{sup} \;\; \iota_{sup}}} \iota_{inf} \Pi_{inf}\Pi_{sub} \\ & \simeq \iota_{sup} \iota_{inf} \Pi_{inf} \Pi_{sub} \\ & \simeq \Re \end{aligned}

(…)

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Basic notions of homotopy theory

Traditionally, mathematics and physics have been founded on set theory, whose concept of sets is that of “bags of distinguishable points”.

But fundamental physics is governed by the gauge principle. This says that given any two “things”, such as two field histories xx and yy, it is in general wrong to ask whether they are equal or not, instead one has to ask where there is a gauge transformation

xγy x \stackrel{\gamma}{\longrightarrow} y

between them. In mathematics this is called a homotopy.

This principle applies also to gauge transformations/homotopies themselves, and thus leads to gauge-of-gauge transformations or homotopies of homotopies

and so on to ever higher gauge transformations or higher homotopies:

This shows that what xx an yy here are elements of is not really a set in the sense of set theory. Instead, such a collection of elements with higher gauge transformations/higher homotopies between them is called a homotopy type.

Hence the theory of homotopy typeshomotopy theory – is much like set theory, but with the concept of gauge transformation/homotopy built right into its foundations. Homotopy theory is gauged mathematics.

A classical model for homotopy types are simply topological spaces: Their points represent the elements, the continuous paths between points represent the gauge transformations, and continuous deformations of paths represent higher gauge transformations. A central result of homotopy theory is the proof of the homotopy hypothesis, which says that under this identification homotopy types are equivalent to topological spaces viewed, in turn, up to “weak homotopy equivalence”.

In the special case of a homotopy type with a single element xx, the gauge transformations necessarily go from xx to itself and hence form a group of symmetries of xx.

This way homotopy theory subsumes group theory.

If there are higher order gauge-of-gauge transformations/homotopies of homotopies between these symmetry group-elements, then one speaks of 2-groups, 3-groups, … n-groups, and eventually of ∞-groups. When homotopy types are represented by topological spaces, then ∞-groups are represented by topological groups.

This way homotopy theory subsumes parts of topological group theory.

Since, generally, there is more than one element in a homotopy type, these are like “groups with several elements”, and as such they are called groupoids (Def. ).

If there are higher order gauge-of-gauge transformations/homotopies of homotopies between the transformations in such a groupoid, one speaks of 2-groupoids, 3-groupoids, … n-groupoids, and eventually of ∞-groupoids. The plain sets are recovered as the special case of 0-groupoids.

Due to the higher orders nn appearing here, mathematical structures based not on sets but on homotopy types are also called higher structures.

Hence homotopy types are equivalently ∞-groupoids. This perspective makes explicit that homotopy types are the unification of plain sets with the concept of gauge-symmetry groups.

An efficient way of handling ∞-groupoids is in their explicit guise as Kan complexes (Def. below); these are the non-abelian generalization of the chain complexes used in homological algebra. Indeed, chain homotopy is a special case of the general concept of homotopy, and hence homological algebra forms but a special abelian corner within homotopy theory. Conversely, homotopy theory may be understood as the non-abelian generalization of homological algebra.

Hence, in a self-reflective manner, there are many different but equivalent incarnations of homotopy theory. Below we discuss in turn:

\,

Ideally, abstract homotopy theory would simply be a complete replacement of set theory, obtained by removing the assumption of strict equality, relaxing it to gauge equivalence/homotopy. As such, abstract homotopy theory would be part and parcel of the foundations of mathematics themselves, not requiring any further discussion. This ideal perspective is the promise of homotopy type theory and may become full practical reality in the next decades.

Until then, abstract homotopy theory has to be formulated on top of the traditional foundations of mathematics provided by set theory, much like one may have to run a Linux emulator on a Windows machine, if one does happen to be stuck with the latter.

A very convenient and powerful such emulator for homotopy theory within set theory is model category theory, originally due to Quillen 67 and highly developed since. This we introduce here.

The idea is to consider ordinary categories (Def. ) but with the understanding that some of their morphisms

XfY X \overset{f}{\longrightarrow} Y

should be homotopy equivalences (Def. ), namely similar to isomorphisms (Def. ), but not necessarily satisfying the two equations defining an actual isomorphism

f 1f=id XAAAAff 1=id Y f^{-1} \circ f \;=\; id_{X} \phantom{AAAA} f \circ f^{-1} \;=\; id_Y

but intended to satisfy this only with equality relaxed to gauge transformation/homotopy:

(92)f 1fgaugeid XAAAAff 1gaugeid Y. f^{-1} \circ f \;\overset{gauge}{\Rightarrow}\; id_{X} \phantom{AAAA} f \circ f^{-1} \;\overset{gauge}{\Rightarrow}\; id_Y \,.

Such would-be homotopy equivalences are called weak equivalences (Def. below).

In principle, this information already defines a homotopy theory by a construction called simplicial localization, which turns weak equivalences into actual homotopy equivalences in a suitable way.

However, without further tools this construction is unwieldy. The extra structure of a model category (Def. below) on top of a category with weak equivalences provides a set of tools.

The idea here is to abstract (in Def. below) from the evident concepts in topological homotopy theory of left homotopy (Def. ) and right homotopy (Def. ) between continuous functions: These are provided by continuous functions out of a cylinder space Cyl(X)=X×[0,1]Cyl(X) = X \times [0,1] or into a path space Path(X)=X [0,1]Path(X) = X^{[0,1]}, respectively, where in both cases the interval space [0,1][0,1] serves to parameterize the relevant gauge transformation/homotopy.

Now a little reflection shows (this was the seminal insight of Quillen 67) that what really matters in this construction of homotopies is that the path space factors the diagonal morphism from a space XX to its Cartesian product as

diag X:Xweak equiv. cofibration Path(X) fibration X×X diag_X \;\colon\; X \underoverset{\text{weak equiv.}}{\text{ cofibration }}{\longrightarrow} Path(X) \overset{\text{ fibration }}{\longrightarrow} X \times X

while the cylinder serves to factor the codiagonal morphism as

codiag X:XXcofibrationCyl(X)weak equivfibrationX codiag_X \;\colon\; X \sqcup X \overset{ \text{cofibration} }{\longrightarrow} Cyl(X) \underoverset{ \text{weak equiv} }{ \text{fibration} }{\longrightarrow} X

where in both cases “fibration” means something like well behaved surjection, while “cofibration” means something like satisfying the lifting property (Def. below) against fibrations that are also weak equivalences.

Such factorizations subject to lifting properties is what the definition of model category axiomatizes, in some generality. That this indeed provides a good toolbox for handling homotopy equivalences is shown by the Whitehead theorem in model categories (Lemma below), which exhibits all weak equivalences as actual homotopy equivalences after passage to “good representatives” of objects (fibrant/cofibrant resolutions, Def. below). Accordingly, the first theorem of model category theory (Quillen 67, I.1 theorem 1, reproduced as Theorem below), provides a tractable expression for the hom-sets modulo homotopy equivalence of the underlying category with weak equivalences in terms of actual morphisms out of cofibrant resolutions into fibrant resolutions (Lemma below).

This is then generally how model category-theory serves as a model for homotopy theory: All homotopy-theoretic constructions, such as that of long homotopy fiber sequences (Prop. below), are reflected via constructions of ordinary category theory but applied to suitably resolved objects.

\,

Literature (Dwyer-Spalinski 95)

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Definition

(model category)

A model category is

  1. a category 𝒞\mathcal{C} (Def. ) with all limits and colimits (Def. );

  2. three sub-classes W,Fib,CofMor(𝒞)W, Fib, Cof \subset Mor(\mathcal{C}) of its class of morphisms;

such that

  1. the class WW makes 𝒞\mathcal{C} into a category with weak equivalences, def. ;

  2. The pairs (WCof,Fib)(W \cap Cof\;,\; Fib) and (Cap,WFib)(Cap\;,\; W\cap Fib) are both weak factorization systems, def. .

One says:

The form of def. is due to (Joyal, def. E.1.2). It implies various other conditions that (Quillen 67) demands explicitly, see prop. and prop. below.

We now dicuss the concept of weak factorization systems (Def. below) appearing in def. .

Factorization systems

Definition

(lift and extension)

Let 𝒞\mathcal{C} be any category. Given a diagram in 𝒞\mathcal{C} of the form

X f Y p B \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow \\ B }

then an extension of the morphism ff along the morphism pp is a completion to a commuting diagram of the form

X f Y p f˜ B. \array{ X &\stackrel{f}{\longrightarrow}& Y \\ {}^{\mathllap{p}}\downarrow & \nearrow_{\mathrlap{\tilde f}} \\ B } \,.

Dually, given a diagram of the form

A p X f Y \array{ && A \\ && \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y }

then a lift of ff through pp is a completion to a commuting diagram of the form

A f˜ p X f Y. \array{ && A \\ &{}^{\mathllap{\tilde f}}\nearrow& \downarrow^{\mathrlap{p}} \\ X &\stackrel{f}{\longrightarrow}& Y } \,.

Combining these cases: given a commuting square

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2 \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow && \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 }

then a lifting in the diagram is a completion to a commuting diagram of the form

X 1 f 1 Y 1 p l p r X 2 f 1 Y 2. \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ {}^{\mathllap{p_l}}\downarrow &\nearrow& \downarrow^{\mathrlap{p_r}} \\ X_2 &\stackrel{f_1}{\longrightarrow}& Y_2 } \,.

Given a sub-class of morphisms KMor(𝒞)K \subset Mor(\mathcal{C}), then

  • a morphism p rp_r as above is said to have the right lifting property against KK or to be a KK-injective morphism if in all square diagrams with p rp_r on the right and any p lKp_l \in K on the left a lift exists.

dually:

  • a morphism p lp_l is said to have the left lifting property against KK or to be a KK-projective morphism if in all square diagrams with p lp_l on the left and any p rKp_r \in K on the left a lift exists.
Definition

(weak factorization systems)

A weak factorization system (WFS) on a category 𝒞\mathcal{C} is a pair (Proj,Inj)(Proj,Inj) of classes of morphisms of 𝒞\mathcal{C} such that

  1. Every morphism f:XYf \colon X\to Y of 𝒞\mathcal{C} may be factored as the composition of a morphism in ProjProj followed by one in InjInj

    f:XProjZInjY. f\;\colon\; X \overset{\in Proj}{\longrightarrow} Z \overset{\in Inj}{\longrightarrow} Y \,.
  2. The classes are closed under having the lifting property, def. , against each other:

    1. ProjProj is precisely the class of morphisms having the left lifting property against every morphisms in InjInj;

    2. InjInj is precisely the class of morphisms having the right lifting property against every morphisms in ProjProj.

Definition

(functorial factorization)

For 𝒞\mathcal{C} a category, a functorial factorization of the morphisms in 𝒞\mathcal{C} is a functor

fact:𝒞 Δ[1]𝒞 Δ[2] fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}

which is a section of the composition functor d 1:𝒞 Δ[2]𝒞 Δ[1]d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}.

Remark

In def. we are using the following standard notation, see at simplex category and at nerve of a category:

Write [1]={01}[1] = \{0 \to 1\} and [2]={012}[2] = \{0 \to 1 \to 2\} for the ordinal numbers, regarded as posets and hence as categories. The arrow category Arr(𝒞)Arr(\mathcal{C}) is equivalently the functor category 𝒞 Δ[1]Funct(Δ[1],𝒞)\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C}), while 𝒞 Δ[2]Funct(Δ[2],𝒞)\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C}) has as objects pairs of composable morphisms in 𝒞\mathcal{C}. There are three injective functors δ i:[1][2]\delta_i \colon [1] \rightarrow [2], where δ i\delta_i omits the index ii in its image. By precomposition, this induces functors d i:𝒞 Δ[2]𝒞 Δ[1]d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}. Here

  • d 1d_1 sends a pair of composable morphisms to their composition;

  • d 2d_2 sends a pair of composable morphisms to the first morphisms;

  • d 0d_0 sends a pair of composable morphisms to the second morphisms.

Definition

A weak factorization system, def. , is called a functorial weak factorization system if the factorization of morphisms may be chosen to be a functorial factorization factfact, def. , i.e. such that d 2factd_2 \circ fact lands in ProjProj and d 0factd_0\circ fact in InjInj.

Remark

Not all weak factorization systems are functorial, def. , although most (including those produced by the small object argument (prop. below), with due care) are.

Proposition

Let 𝒞\mathcal{C} be a category and let KMor(𝒞)K\subset Mor(\mathcal{C}) be a class of morphisms. Write KProjK Proj and KInjK Inj, respectively, for the sub-classes of KK-projective morphisms and of KK-injective morphisms, def. . Then:

  1. Both classes contain the class of isomorphism of 𝒞\mathcal{C}.

  2. Both classes are closed under composition in 𝒞\mathcal{C}.

    KProjK Proj is also closed under transfinite composition.

  3. Both classes are closed under forming retracts in the arrow category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} (see remark ).

  4. KProjK Proj is closed under forming pushouts of morphisms in 𝒞\mathcal{C} (“cobase change”).

    KInjK Inj is closed under forming pullback of morphisms in 𝒞\mathcal{C} (“base change”).

  5. KProjK Proj is closed under forming coproducts in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}.

    KInjK Inj is closed under forming products in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}.

Proof

We go through each item in turn.

containing isomorphisms

Given a commuting square

A f X Iso i p B g Y \array{ A &\overset{f}{\rightarrow}& X \\ {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}} \\ B &\underset{g}{\longrightarrow}& Y }

with the left morphism an isomorphism, then a lift is given by using the inverse of this isomorphism fi 1{}^{{f \circ i^{-1}}}\nearrow. Hence in particular there is a lift when pKp \in K and so iKProji \in K Proj. The other case is formally dual.

closure under composition

Given a commuting square of the form

A X KInj p 1 K i KInj p 2 B Y \array{ A &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }

consider its pasting decomposition as

A X KInj p 1 K i KInj p 2 B Y. \array{ A &\longrightarrow& X \\ \downarrow &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y } \,.

Now the bottom commuting square has a lift, by assumption. This yields another pasting decomposition

A X K i KInj p 1 KInj p 2 B Y \array{ A &\longrightarrow& X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}} \\ \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}} \\ B &\longrightarrow& Y }

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that p 1p 1p_1\circ p_1 has the right lifting property against KK and is hence in KInjK Inj. The case of composing two morphisms in KProjK Proj is formally dual. From this the closure of KProjK Proj under transfinite composition follows since the latter is given by colimits of sequential composition and successive lifts against the underlying sequence as above constitutes a cocone, whence the extension of the lift to the colimit follows by its universal property.

closure under retracts

Let jj be the retract of an iKProji \in K Proj, i.e. let there be a commuting diagram of the form.

id A: A C A j KProj i j id B: B D B. \array{ id_A \colon & A &\longrightarrow& C &\longrightarrow& A \\ & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} \\ id_B \colon & B &\longrightarrow& D &\longrightarrow& B } \,.

Then for

A X j K f B Y \array{ A &\longrightarrow& X \\ {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& Y }

a commuting square, it is equivalent to its pasting composite with that retract diagram

A C A X j KProj i j K f B D B Y. \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.

Here the pasting composite of the two squares on the right has a lift, by assumption:

A C A X j i K f B D B Y. \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{} && \nearrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,.

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence jj has the left lifting property against all pKp \in K and hence is in KProjK Proj. The other case is formally dual.

closure under pushout and pullback

Let pKInjp \in K Inj and and let

Z× fX X f *p p Z f Y \array{ Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}} \\ Z &\stackrel{f}{\longrightarrow} & Y }

be a pullback diagram in 𝒞\mathcal{C}. We need to show that f *pf^* p has the right lifting property with respect to all iKi \in K. So let

A Z× fX K i f *p B g Z \array{ A &\longrightarrow& Z \times_f X \\ {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}} \\ B &\stackrel{g}{\longrightarrow}& Z }

be a commuting square. We need to construct a diagonal lift of that square. To that end, first consider the pasting composite with the pullback square from above to obtain the commuting diagram

A Z× fX X i f *p p B g Z f Y. \array{ A &\longrightarrow& Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.

By the right lifting property of pp, there is a diagonal lift of the total outer diagram

A X i (fg)^ p B fg Y. \array{ A &\longrightarrow& X \\ \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{f g}{\longrightarrow}& Y } \,.

By the universal property of the pullback this gives rise to the lift g^\hat g in

Z× fX X g^ f *p p B g Z f Y. \array{ && Z \times_f X &\longrightarrow& X \\ &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,.

In order for g^\hat g to qualify as the intended lift of the total diagram, it remains to show that

A Z× fX i g^ B \array{ A &\longrightarrow& Z \times_f X \\ \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow \\ B }

commutes. To do so we notice that we obtain two cones with tip AA:

  • one is given by the morphisms

    1. AZ× fXXA \to Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z

    with universal morphism into the pullback being

    • AZ× fXA \to Z \times_f X
  • the other by

    1. AiBg^Z× fXXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X
    2. AiBgZA \stackrel{i}{\to} B \stackrel{g}{\to} Z.

    with universal morphism into the pullback being

    • AiBg^Z× fXA \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is unique this implies the required identity of morphisms.

The other case is formally dual.

closure under (co-)products

Let {(A si sB s)KProj} sS\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S} be a set of elements of KProjK Proj. Since colimits in the presheaf category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} are computed componentwise, their coproduct in this arrow category is the universal morphism out of the coproduct of objects sSA s\underset{s \in S}{\coprod} A_s induced via its universal property by the set of morphisms i si_s:

sSA s(i s) sSsSB s. \underset{s \in S}{\sqcup} A_s \overset{(i_s)_{s\in S}}{\longrightarrow} \underset{s \in S}{\sqcup} B_s \,.

Now let

sSA s X (i s) sS K f sSB s Y \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y }

be a commuting square. This is in particular a cocone under the coproduct of objects, hence by the universal property of the coproduct, this is equivalent to a set of commuting diagrams

{A s X KProj i s K f B s Y} sS. \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in K Proj}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} \,.

By assumption, each of these has a lift s\ell_s. The collection of these lifts

{A s X Proj i s s K f B s Y} sS \left\{ \;\;\;\;\;\;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in Proj}}\downarrow &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S}

is now itself a compatible cocone, and so once more by the universal property of the coproduct, this is equivalent to a lift ( s) sS(\ell_s)_{s\in S} in the original square

sSA s X (i s) sS ( s) sS K f sSB s Y. \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow &{}^{(\ell_s)_{s\in S}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } \,.

This shows that the coproduct of the i si_s has the left lifting property against all fKf\in K and is hence in KProjK Proj. The other case is formally dual.

An immediate consequence of prop. is this:

Corollary

Let 𝒞\mathcal{C} be a category with all small colimits, and let KMor(𝒞)K\subset Mor(\mathcal{C}) be a sub-class of its morphisms. Then every KK-injective morphism, def. , has the right lifting property, def. , against all KK-relative cell complexes, def. and their retracts, remark .

Remark

By a retract of a morphism XfYX \stackrel{f}{\longrightarrow} Y in some category 𝒞\mathcal{C} we mean a retract of ff as an object in the arrow category 𝒞 Δ[1]\mathcal{C}^{\Delta[1]}, hence a morphism AgBA \stackrel{g}{\longrightarrow} B such that in 𝒞 Δ[1]\mathcal{C}^{\Delta[1]} there is a factorization of the identity on gg through ff

id g:gfg. id_g \;\colon\; g \longrightarrow f \longrightarrow g \,.

This means equivalently that in 𝒞\mathcal{C} there is a commuting diagram of the form

id A: A X A g f g id B: B Y B. \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,.
Lemma

In every category CC the class of isomorphisms is preserved under retracts in the sense of remark .

Proof

For

id A: A X A g f g id B: B Y B. \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,.

a retract diagram and XfYX \overset{f}{\to} Y an isomorphism, the inverse to AgBA \overset{g}{\to} B is given by the composite

X A f 1 B Y . \array{ & & & X & \longrightarrow & A \\ & && \uparrow^{\mathrlap{f^{-1}}} && \\ & B & \longrightarrow& Y&& } \,.

More generally:

Proposition

Given a model category in the sense of def. , then its class of weak equivalences is closed under forming retracts (in the arrow category, see remark ).

(Joyal, prop. E.1.3)

Proof

Let

id: A X A f w f id: B Y B \array{ id \colon & A &\longrightarrow& X &\longrightarrow& A \\ & {}^{\mathllap{f}} \downarrow && \downarrow^{\mathrlap{w}} && \downarrow^{\mathrlap{f}} \\ id \colon & B &\longrightarrow& Y &\longrightarrow& B }

be a commuting diagram in the given model category, with wWw \in W a weak equivalence. We need to show that then also fWf \in W.

First consider the case that fFibf \in Fib.

In this case, factor ww as a cofibration followed by an acyclic fibration. Since wWw \in W and by two-out-of-three (def. ) this is even a factorization through an acyclic cofibration followed by an acyclic fibration. Hence we obtain a commuting diagram of the following form:

id: A X AAAA A id WCof id id: A s X AAtAA A Fib f WFib Fib f id: B Y AAAA B, \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{id}} \\ id \colon & A' &\overset{s}{\longrightarrow}& X' &\overset{\phantom{AA}t\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{f}}_{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,

where ss is uniquely defined and where tt is any lift of the top middle vertical acyclic cofibration against ff. This now exhibits ff as a retract of an acyclic fibration. These are closed under retract by prop. .

Now consider the general case. Factor ff as an acyclic cofibration followed by a fibration and form the pushout in the top left square of the following diagram

id: A X AAAA A WCof (po) WCof WCof id: A X AAAA A Fib W Fib id: B Y AAAA B, \array{ id \colon & A &\longrightarrow& X &\overset{\phantom{AAAA}}{\longrightarrow}& A \\ & {}^{\mathllap{\in W \cap Cof}}\downarrow &(po)& \downarrow^{\mathrlap{\in W \cap Cof}} && \downarrow^{\mathrlap{\in W \cap Cof}} \\ id \colon & A' &\overset{}{\longrightarrow}& X' &\overset{\phantom{AA}\phantom{AA}}{\longrightarrow}& A' \\ & {}^{\mathllap{\in Fib}} \downarrow && \downarrow^{\mathrlap{\in W }} && \downarrow^{\mathrlap{\in Fib}} \\ id \colon & B &\longrightarrow& Y &\underset{\phantom{AAAA}}{\longrightarrow}& B } \,,

where the other three squares are induced by the universal property of the pushout, as is the identification of the middle horizontal composite as the identity on AA'. Since acyclic cofibrations are closed under forming pushouts by prop. , the top middle vertical morphism is now an acyclic fibration, and hence by assumption and by two-out-of-three so is the middle bottom vertical morphism.

Thus the previous case now gives that the bottom left vertical morphism is a weak equivalence, and hence the total left vertical composite is.

Lemma

(retract argument)

Consider a composite morphism

f:XiApY. f \;\colon\; X \stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y \,.
  1. If ff has the left lifting property against pp, then ff is a retract of ii.

  2. If ff has the right lifting property against ii, then ff is a retract of pp.

Proof

We discuss the first statement, the second is formally dual.

Write the factorization of ff as a commuting square of the form

X i A f p Y = Y. \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,.

By the assumed lifting property of ff against pp there exists a diagonal filler gg making a commuting diagram of the form

X i A f g p Y = Y. \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,.

By rearranging this diagram a little, it is equivalent to

X = X f i id Y: Y g A p Y. \array{ & X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,.

Completing this to the right, this yields a diagram exhibiting the required retract according to remark :

id X: X = X = X f i f id Y: Y g A p Y. \array{ id_X \colon & X &=& X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow && {}^{\mathllap{f}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,.

Small object argument

Given a set CMor(𝒞)C \subset Mor(\mathcal{C}) of morphisms in some category 𝒞\mathcal{C}, a natural question is how to factor any given morphism f:XYf\colon X \longrightarrow Y through a relative CC-cell complex, def. , followed by a CC-injective morphism, def.

f:XCcellX^CinjY. f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,.

A first approximation to such a factorization turns out to be given simply by forming X^=X 1\hat X = X_1 by attaching all possible CC-cells to XX. Namely let

(C/f){dom(c) X cC f cod(c) Y} (C/f) \coloneqq \left\{ \array{ dom(c) &\stackrel{}{\longrightarrow}& X \\ {}^{\mathllap{c\in C}}\downarrow && \downarrow^{\mathrlap{f}} \\ cod(c) &\longrightarrow& Y } \right\}

be the set of all ways to find a CC-cell attachment in ff, and consider the pushout X^\hat X of the coproduct of morphisms in CC over all these:

c(C/f)dom(c) X c(C/f)c (po) c(C/f)cod(c) X 1. \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &(po)& \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& X_1 } \,.

This gets already close to producing the intended factorization:

First of all the resulting map XX 1X \to X_1 is a CC-relative cell complex, by construction.

Second, by the fact that the coproduct is over all commuting squres to ff, the morphism ff itself makes a commuting diagram

c(C/f)dom(c) X c(C/f)c f c(C/f)cod(c) Y \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow && \downarrow^{\mathrlap{f}} \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y }

and hence the universal property of the colimit means that ff is indeed factored through that CC-cell complex X 1X_1; we may suggestively arrange that factorizing diagram like so:

c(C/f)dom(c) X id c(C/f)dom(c) X 1 c(C/f)c c(C/f)cod(c) Y. \array{ \underset{c\in(C/f)}{\coprod} dom(c) &\longrightarrow& X \\ {}^{\mathllap{id}}\downarrow && \downarrow^{\mathrlap{}} \\ \underset{c\in(C/f)}{\coprod} dom(c) && X_1 \\ {}^{\mathllap{\underset{c\in(C/f)}{\coprod} c}}\downarrow &\nearrow& \downarrow \\ \underset{c\in(C/f)}{\coprod} cod(c) &\longrightarrow& Y } \,.

This shows that, finally, the colimiting co-cone map – the one that now appears diagonally – almost exhibits the desired right lifting of X 1YX_1 \to Y against the cCc\in C. The failure of that to hold on the nose is only the fact that a horizontal map in the middle of the above diagram is missing: the diagonal map obtained above lifts not all commuting diagrams of cCc\in C into ff, but only those where the top morphism dom(c)X 1dom(c) \to X_1 factors through XX 1X \to X_1.

The idea of the small object argument now is to fix this only remaining problem by iterating the construction: next factor X 1YX_1 \to Y in the same way into

X 1X 2Y X_1 \longrightarrow X_2 \longrightarrow Y

and so forth. Since relative CC-cell complexes are closed under composition, at stage nn the resulting XX nX \longrightarrow X_n is still a CC-cell complex, getting bigger and bigger. But accordingly, the failure of the accompanying X nYX_n \longrightarrow Y to be a CC-injective morphism becomes smaller and smaller, for it now lifts against all diagrams where dom(c)X ndom(c) \longrightarrow X_n factors through X n1X nX_{n-1}\longrightarrow X_n, which intuitively is less and less of a condition as the X n1X_{n-1} grow larger and larger.

The concept of small object is just what makes this intuition precise and finishes the small object argument. For the present purpose we just need the following simple version:

Definition

For 𝒞\mathcal{C} a category and CMor(𝒞)C \subset Mor(\mathcal{C}) a sub-set of its morphisms, say that these have small domains if there is an ordinal α\alpha (def. ) such that for every cCc\in C and for every CC-relative cell complex given by a transfinite composition (def. )

f:XX 1X 2X βX^ f \;\colon\; X \to X_1 \to X_2 \to \cdots \to X_\beta \to \cdots \longrightarrow \hat X

every morphism dom(c)X^dom(c)\longrightarrow \hat X factors through a stage X βX^X_\beta \to \hat X of order β<α\beta \lt \alpha:

X β dom(c) X^. \array{ && X_\beta \\ & \nearrow & \downarrow \\ dom(c) &\longrightarrow& \hat X } \,.

The above discussion proves the following:

Proposition

(small object argument)

Let 𝒞\mathcal{C} be a locally small category with all small colimits. If a set CMor(𝒞)C\subset Mor(\mathcal{C}) of morphisms has all small domains in the sense of def. , then every morphism f:Xf\colon X\longrightarrow in 𝒞\mathcal{C} factors through a CC-relative cell complex, def. , followed by a CC-injective morphism, def.

f:XCcellX^CinjY. f \;\colon\; X \stackrel{\in C cell}{\longrightarrow} \hat X \stackrel{\in C inj}{\longrightarrow} Y \,.

(Quillen 67, II.3 lemma)

Homotopy

We discuss how the concept of homotopy is abstractly realized in model categories, def. .

Definition

Let 𝒞\mathcal{C} be a model category, def. , and X𝒞X \in \mathcal{C} an object.

  • A path space object Path(X)Path(X) for XX is a factorization of the diagonal Δ X:XX×X\Delta_X \;\colon\; X \to X \times X as
Δ X:XWiPath(X)Fib(p 0,p 1)X×X. \Delta_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \underoverset{\in Fib}{(p_0,p_1)}{\longrightarrow} X \times X \,.

where XPath(X)X\to Path(X) is a weak equivalence and Path(X)X×XPath(X) \to X \times X is a fibration.

  • A cylinder object Cyl(X)Cyl(X) for XX is a factorization of the codiagonal (or “fold map”) X:XXX\nabla_X \;\colon\; X \sqcup X \to X as
X:XXCof(i 0,i 1)Cyl(X)WpX. \nabla_X \;\colon\; X \sqcup X \underoverset{\in Cof}{(i_0,i_1)}{\longrightarrow} Cyl(X) \underoverset{\in W}{p}{\longrightarrow} X \,.

where Cyl(X)XCyl(X) \to X is a weak equivalence. and XXCyl(X)X \sqcup X \to Cyl(X) is a cofibration.

Remark

For every object X𝒞X \in \mathcal{C} in a model category, a cylinder object and a path space object according to def. exist: the factorization axioms guarantee that there exists

  1. a factorization of the codiagonal as

    X:XXCofCyl(X)WFibX \nabla_X \;\colon\; X \sqcup X \overset{\in Cof}{\longrightarrow} Cyl(X) \overset{\in W \cap Fib}{\longrightarrow} X
  2. a factorization of the diagonal as

    Δ X:XWCofPath(X)FibX×X. \Delta_X \;\colon\; X \overset{\in W \cap Cof}{\longrightarrow} Path(X) \overset{\in Fib}{\longrightarrow} X \times X \,.

The cylinder and path space objects obtained this way are actually better than required by def. : in addition to Cyl(X)XCyl(X)\to X being just a weak equivalence, for these this is actually an acyclic fibration, and dually in addition to XPath(X)X\to Path(X) being a weak equivalence, for these it is actually an acyclic cofibrations.

Some authors call cylinder/path-space objects with this extra property “very good” cylinder/path-space objects, respectively.

One may also consider dropping a condition in def. : what mainly matters is the weak equivalence, hence some authors take cylinder/path-space objects to be defined as in def. but without the condition that XXCyl(X)X \sqcup X\to Cyl(X) is a cofibration and without the condition that Path(X)XPath(X) \to X is a fibration. Such authors would then refer to the concept in def. as “good” cylinder/path-space objects.

The terminology in def. follows the original (Quillen 67, I.1 def. 4). With the induced concept of left/right homotopy below in def. , this admits a quick derivation of the key facts in the following, as we spell out below.

Lemma

Let 𝒞\mathcal{C} be a model category. If X𝒞X \in \mathcal{C} is cofibrant, then for every cylinder object Cyl(X)Cyl(X) of XX, def. , not only is (i 0,i 1):XXX(i_0,i_1) \colon X \sqcup X \to X a cofibration, but each

i 0,i 1:XCyl(X) i_0, i_1 \colon X \longrightarrow Cyl(X)

is an acyclic cofibration separately.

Dually, if X𝒞X \in \mathcal{C} is fibrant, then for every path space object Path(X)Path(X) of XX, def. , not only is (p 0,p 1):Path(X)X×X(p_0,p_1) \colon Path(X)\to X \times X a cofibration, but each

p 0,p 1:Path(X)X p_0, p_1 \colon Path(X) \longrightarrow X

is an acyclic fibration separately.

Proof

We discuss the case of the path space object. The other case is formally dual.

First, that the component maps are weak equivalences follows generally: by definition they have a right inverse Path(X)XPath(X) \to X and so this follows by two-out-of-three (def. ).

But if XX is fibrant, then also the two projection maps out of the product X×XXX \times X \to X are fibrations, because they are both pullbacks of the fibration X*X \to \ast

X×X X (pb) X *. \array{ X\times X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ X &\longrightarrow& \ast } \,.

hence p i:Path(X)X×XXp_i \colon Path(X)\to X \times X \to X is the composite of two fibrations, and hence itself a fibration, by prop. .

Path space objects are very non-unique as objects up to isomorphism:

Example

If X𝒞X \in \mathcal{C} is a fibrant object in a model category, def. , and for Path 1(X)Path_1(X) and Path 2(X)Path_2(X) two path space objects for XX, def. , then the fiber product Path 1(X)× XPath 2(X)Path_1(X) \times_X Path_2(X) is another path space object for XX: the pullback square

X Δ X X×X Path 1(X)×XPath 2(X) Path 1(X)×Path 2(X) Fib (pb) Fib X×X×X (id,Δ X,id) X×X×X×X Fib (pr 1,pr 3) (p 1,p 4) X×X = X×X \array{ X &\overset{\Delta_X}{\longrightarrow}& X \times X \\ \downarrow && \downarrow \\ Path_1(X) \underset{X}{\times} Path_2(X) &\longrightarrow& Path_1(X)\times Path_2(X) \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ X \times X \times X &\overset{(id,\Delta_X,id)}{\longrightarrow}& X \times X\times X \times X \\ \downarrow^{\mathrlap{(pr_1,pr_3)}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{(p_1, p_4)}} \\ X\times X &=& X \times X }

gives that the induced projection is again a fibration. Moreover, using lemma and two-out-of-three (def. ) gives that XPath 1(X)× XPath 2(X)X \to Path_1(X) \times_X Path_2(X) is a weak equivalence.

For the case of the canonical topological path space objects of def , with Path 1(X)=Path 2(X)=X I=X [0,1]Path_1(X) = Path_2(X) = X^I = X^{[0,1]} then this new path space object is X II=X [0,2]X^{I \vee I} = X^{[0,2]}, the mapping space out of the standard interval of length 2 instead of length 1.

Definition

(abstract left homotopy and abstract right homotopy

Let f,g:XYf,g \colon X \longrightarrow Y be two parallel morphisms in a model category.

  • A left homotopy η:f Lg\eta \colon f \Rightarrow_L g is a morphism η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y from a cylinder object of XX, def. , such that it makes this diagram commute:
X Cyl(X) X f η g Y. \array{ X &\longrightarrow& Cyl(X) &\longleftarrow& X \\ & {}_{\mathllap{f}}\searrow &\downarrow^{\mathrlap{\eta}}& \swarrow_{\mathrlap{g}} \\ && Y } \,.
  • A right homotopy η:f Rg\eta \colon f \Rightarrow_R g is a morphism η:XPath(Y)\eta \colon X \to Path(Y) to some path space object of XX, def. , such that this diagram commutes:
X f η g Y Path(Y) Y. \array{ && X \\ & {}^{\mathllap{f}}\swarrow & \downarrow^{\mathrlap{\eta}} & \searrow^{\mathrlap{g}} \\ Y &\longleftarrow& Path(Y) &\longrightarrow& Y } \,.
Lemma

Let f,g:XYf,g \colon X \to Y be two parallel morphisms in a model category.

  1. Let XX be cofibrant. If there is a left homotopy f Lgf \Rightarrow_L g then there is also a right homotopy f Rgf \Rightarrow_R g (def. ) with respect to any chosen path space object.

  2. Let XX be fibrant. If there is a right homotopy f Rgf \Rightarrow_R g then there is also a left homotopy f Lgf \Rightarrow_L g with respect to any chosen cylinder object.

In particular if XX is cofibrant and YY is fibrant, then by going back and forth it follows that every left homotopy is exhibited by every cylinder object, and every right homotopy is exhibited by every path space object.

Proof

We discuss the first case, the second is formally dual. Let η:Cyl(X)Y\eta \colon Cyl(X) \longrightarrow Y be the given left homotopy. Lemma implies that we have a lift hh in the following commuting diagram

X if Path(Y) WCof i 0 h Fib p 0,p 1 Cyl(X) (fp,η) Y×Y, \array{ X &\overset{i \circ f}{\longrightarrow}& Path(Y) \\ {}^{\mathllap{i_0}}_{\mathllap{\in W \cap Cof}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,,

where on the right we have the chosen path space object. Now the composite η˜hi 1\tilde \eta \coloneqq h \circ i_1 is a right homotopy as required:

Path(Y) h Fib p 0,p 1 X i 1 Cyl(X) (fp,η) Y×Y. \array{ && && Path(Y) \\ && &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p_0,p_1}}_{\mathrlap{\in Fib}} \\ X &\overset{i_1}{\longrightarrow}& Cyl(X) &\underset{(f \circ p,\eta)}{\longrightarrow}& Y \times Y } \,.
Proposition

For XX a cofibrant object in a model category and YY a fibrant object, then the relations of left homotopy f Lgf \Rightarrow_L g and of right homotopy f Rgf \Rightarrow_R g (def. ) on the hom set Hom(X,Y)Hom(X,Y) coincide and are both equivalence relations.

Proof

That both relations coincide under the (co-)fibrancy assumption follows directly from lemma .

The symmetry and reflexivity of the relation is obvious.

That right homotopy (hence also left homotopy) with domain XX is a transitive relation follows from using example to compose path space objects.

The homotopy category

We discuss the construction that takes a model category, def. , and then universally forces all its weak equivalences into actual isomorphisms.

Definition

(homotopy category of a model category)

Let 𝒞\mathcal{C} be a model category, def. . Write Ho(𝒞)Ho(\mathcal{C}) for the category whose

and whose composition operation is given on representatives by composition in 𝒞\mathcal{C}.

This is, up to equivalence of categories, the homotopy category of the model category 𝒞\mathcal{C}.

Proposition

Def. is well defined, in that composition of morphisms between fibrant-cofibrant objects in 𝒞\mathcal{C} indeed passes to homotopy classes.

Proof

Fix any morphism XfYX \overset{f}{\to} Y between fibrant-cofibrant objects. Then for precomposition

()[f]:Hom Ho(𝒞)(Y,Z)Hom Ho(𝒞(X,Z)) (-) \circ [f] \;\colon\; Hom_{Ho(\mathcal{C})}(Y,Z) \to Hom_{Ho(\mathcal{C}(X,Z))}

to be well defined, we need that with (gh):YZ(g\sim h)\;\colon\; Y \to Z also (fgfh):XZ(f g \sim f h)\;\colon\; X \to Z. But by prop we may take the homotopy \sim to be exhibited by a right homotopy η:YPath(Z)\eta \colon Y \to Path(Z), for which case the statement is evident from this diagram:

Z g p 1 X f Y η Path(Z) h p 0 Z. \array{ && && Z \\ && & {}^{\mathllap{g}}\nearrow & \uparrow^{\mathrlap{p_1}} \\ X &\overset{f}{\longrightarrow} & Y &\overset{\eta}{\longrightarrow}& Path(Z) \\ && & {}_{\mathllap{h}}\searrow & \downarrow_{\mathrlap{p_0}} \\ && && Z } \,.

For postcomposition we may choose to exhibit homotopy by left homotopy and argue dually.

We now spell out that def. indeed satisfies the universal property that defines the localization of a category with weak equivalences at its weak equivalences.

Lemma

(Whitehead theorem in model categories)

Let 𝒞\mathcal{C} be a model category. A weak equivalence between two objects which are both fibrant and cofibrant is a homotopy equivalence (92).

Proof

By the factorization axioms in the model category 𝒞\mathcal{C} and by two-out-of-three (def. ), every weak equivalence f:XYf\colon X \longrightarrow Y factors through an object ZZ as an acyclic cofibration followed by an acyclic fibration. In particular it follows that with XX and YY both fibrant and cofibrant, so is ZZ, and hence it is sufficient to prove that acyclic (co-)fibrations between such objects are homotopy equivalences.

So let f:XYf \colon X \longrightarrow Y be an acyclic fibration between fibrant-cofibrant objects, the case of acyclic cofibrations is formally dual. Then in fact it has a genuine right inverse given by a lift f 1f^{-1} in the diagram

X cof f 1 FibW f X = X. \array{ \emptyset &\rightarrow& X \\ {}^{\mathllap{\in cof}}\downarrow &{}^{{f^{-1}}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in Fib \cap W}} \\ X &=& X } \,.

To see that f 1f^{-1} is also a left inverse up to left homotopy, let Cyl(X)Cyl(X) be any cylinder object on XX (def. ), hence a factorization of the codiagonal on XX as a cofibration followed by a an acyclic fibration

XXι XCyl(X)pX X \sqcup X \stackrel{\iota_X}{\longrightarrow} Cyl(X) \stackrel{p}{\longrightarrow} X

and consider the commuting square

XX (f 1f,id) X ι X Cof WFib f Cyl(X) fp Y, \array{ X \sqcup X &\stackrel{(f^{-1}\circ f, id)}{\longrightarrow}& X \\ {}^{\mathllap{\iota_X}}{}_{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{f\circ p}{\longrightarrow}& Y } \,,

which commutes due to f 1f^{-1} being a genuine right inverse of ff. By construction, this commuting square now admits a lift η\eta, and that constitutes a left homotopy η:f 1f Lid\eta \colon f^{-1}\circ f \Rightarrow_L id.

Definition

(fibrant resolution and cofibrant resolution)

Given a model category 𝒞\mathcal{C}, consider a choice for each object X𝒞X \in \mathcal{C} of

  1. a factorization

    ACofAi XQXWFibp XX \emptyset \underoverset{\phantom{A}\in Cof\phantom{A}}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_X}{\longrightarrow} X

    of the initial morphism (Def. ), such that when XX is already cofibrant then p X=id Xp_X = id_X;

  2. a factorization

    XWCofj XPXAFibAq X* X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} P X \underoverset{\phantom{A} \in Fib \phantom{A}}{q_X}{\longrightarrow} \ast

    of the terminal morphism (Def. ), such that when XX is already fibrant then j X=id Xj_X = id_X.

Write then

γ P,Q:𝒞Ho(𝒞) \gamma_{P,Q} \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

for the functor to the homotopy category, def. , which sends an object XX to the object PQXP Q X and sends a morphism f:XYf \colon X \longrightarrow Y to the homotopy class of the result of first lifting in

QY i X Qf p Y QX fp X Y \array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{i_X}}\downarrow &{}^{Q f}\nearrow& \downarrow^{\mathrlap{p_Y}} \\ Q X &\underset{f\circ p_X}{\longrightarrow}& Y }

and then lifting (here: extending) in

QX j QYQf PQY j QX PQf q QY PQX *. \array{ Q X &\overset{j_{Q Y} \circ Q f}{\longrightarrow}& P Q Y \\ {}^{\mathllap{j_{Q X}}}\downarrow &{}^{P Q f}\nearrow& \downarrow^{\mathrlap{q_{Q Y}}} \\ P Q X &\longrightarrow& \ast } \,.
Lemma

The construction in def. is indeed well defined.

Proof

First of all, the object PQXP Q X is indeed both fibrant and cofibrant (as well as related by a zig-zag of weak equivalences to XX):

Cof Cof QX WCof PQX Fib * W X. \array{ \emptyset \\ {}^{\mathllap{\in Cof}}\downarrow & \searrow^{\mathrlap{\in Cof}} \\ Q X &\underset{\in W \cap Cof}{\longrightarrow}& P Q X &\underset{\in Fib}{\longrightarrow}& \ast \\ {}^{\mathllap{\in W}}\downarrow \\ X } \,.

Now to see that the image on morphisms is well defined. First observe that any two choices (Qf) i(Q f)_{i} of the first lift in the definition are left homotopic to each other, exhibited by lifting in

QXQX ((Qf) 1,(Qf) 2) QY Cof WFib p Y Cyl(QX) fp Xσ QX Y. \array{ Q X \sqcup Q X &\stackrel{((Q f)_1, (Q f)_2 )}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_{Y}}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(Q X) &\underset{f \circ p_{X} \circ \sigma_{Q X}}{\longrightarrow}& Y } \,.

Hence also the composites j QY(Q f) ij_{Q Y}\circ (Q_f)_i are left homotopic to each other, and since their domain is cofibrant, then by lemma they are also right homotopic by a right homotopy κ\kappa. This implies finally, by lifting in

QX κ Path(PQY) WCof Fib PQX (R(Qf) 1,P(Qf) 2) PQY×PQY \array{ Q X &\overset{\kappa}{\longrightarrow}& Path(P Q Y) \\ {}^{\mathllap{\in W \cap Cof}}\downarrow && \downarrow^{\mathrlap{\in Fib}} \\ P Q X &\underset{(R (Q f)_1, P (Q f)_2)}{\longrightarrow}& P Q Y \times P Q Y }

that also P(Qf) 1P (Q f)_1 and P(Qf) 2P (Q f)_2 are right homotopic, hence that indeed PQfP Q f represents a well-defined homotopy class.

Finally to see that the assignment is indeed functorial, observe that the commutativity of the lifting diagrams for QfQ f and PQfP Q f imply that also the following diagram commutes

X p X QX j QX PQX f Qf PQf Y p y QY j QY PQY. \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y } \,.

Now from the pasting composite

X p X QX j QX PQX f Qf PQf Y p Y QY j QY PQY g Qg PQg Z p Z QZ j QZ PQZ \array{ X &\overset{p_X}{\longleftarrow}& Q X &\overset{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underset{p_Y}{\longleftarrow}& Q Y &\underset{j_{Q Y}}{\longrightarrow}& P Q Y \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{Q g}} && \downarrow^{\mathrlap{P Q g}} \\ Z &\underset{p_Z}{\longleftarrow}& Q Z &\underset{j_{Q Z}}{\longrightarrow}& P Q Z }

one sees that (PQg)(PQf)(P Q g)\circ (P Q f) is a lift of gfg \circ f and hence the same argument as above gives that it is homotopic to the chosen PQ(gf)P Q(g \circ f).

For the following, recall the concept of natural isomorphism between functors: for F,G:𝒞𝒟F, G \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} two functors, then a natural transformation η:FG\eta \colon F \Rightarrow G is for each object cObj(𝒞)c \in Obj(\mathcal{C}) a morphism η c:F(c)G(c)\eta_c \colon F(c) \longrightarrow G(c) in 𝒟\mathcal{D}, such that for each morphism f:c 1c 2f \colon c_1 \to c_2 in 𝒞\mathcal{C} the following is a commuting square:

F(c 1) η c 1 G(c 1) F(f) G(f) F(c 2) η c 2 G(c 2). \array{ F(c_1) &\overset{\eta_{c_1}}{\longrightarrow}& G(c_1) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{G(f)}} \\ F(c_2) &\underset{\eta_{c_2}}{\longrightarrow}& G(c_2) } \,.

Such η\eta is called a natural isomorphism if its η c\eta_c are isomorphisms for all objects cc.

Definition

(localization of a category category with weak equivalences)

For 𝒞\mathcal{C} a category with weak equivalences, its localization at the weak equivalences is, if it exists,

  1. a category denoted 𝒞[W 1]\mathcal{C}[W^{-1}]

  2. a functor

    γ:𝒞𝒞[W 1] \gamma \;\colon\; \mathcal{C} \longrightarrow \mathcal{C}[W^{-1}]

such that

  1. γ\gamma sends weak equivalences to isomorphisms;

  2. γ\gamma is universal with this property, in that:

    for F:𝒞DF \colon \mathcal{C} \longrightarrow D any functor out of 𝒞\mathcal{C} into any category DD, such that FF takes weak equivalences to isomorphisms, it factors through γ\gamma up to a natural isomorphism ρ\rho

    𝒞 F D γ ρ F˜ Ho(𝒞) \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }

    and this factorization is unique up to unique isomorphism, in that for (F˜ 1,ρ 1)(\tilde F_1, \rho_1) and (F˜ 2,ρ 2)(\tilde F_2, \rho_2) two such factorizations, then there is a unique natural isomorphism κ:F˜ 1F˜ 2\kappa \colon \tilde F_1 \Rightarrow \tilde F_2 making the evident diagram of natural isomorphisms commute.

Theorem

(convenient localization of model categories)

For 𝒞\mathcal{C} a model category, the functor γ P,Q\gamma_{P,Q} in def. (for any choice of PP and QQ) exhibits Ho(𝒞)Ho(\mathcal{C}) as indeed being the localization of the underlying category with weak equivalences at its weak equivalences, in the sense of def. :

𝒞 = 𝒞 γ P,Q γ Ho(𝒞) 𝒞[W 1]. \array{ \mathcal{C} &=& \mathcal{C} \\ {}^{\mathllap{\gamma_{P,Q}}}\downarrow && \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}) &\simeq& \mathcal{C}[W^{-1}] } \,.

(Quillen 67, I.1 theorem 1)

Proof

First, to see that that γ P,Q\gamma_{P,Q} indeed takes weak equivalences to isomorphisms: By two-out-of-three (def. ) applied to the commuting diagrams shown in the proof of lemma , the morphism PQfP Q f is a weak equivalence if ff is:

X p X QX j QX PQX f Qf PQf Y p y QY j QY PQY \array{ X &\underoverset{\simeq}{p_X}{\longleftarrow}& Q X &\underoverset{\simeq}{j_{Q X}}{\longrightarrow}& P Q X \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{Q f}} && \downarrow^{\mathrlap{P Q f}} \\ Y &\underoverset{p_y}{\simeq}{\longleftarrow}& Q Y &\underoverset{j_{Q Y}}{\simeq}{\longrightarrow}& P Q Y }

With this the “Whitehead theorem for model categories”, lemma , implies that PQfP Q f represents an isomorphism in Ho(𝒞)Ho(\mathcal{C}).

Now let F:𝒞DF \colon \mathcal{C}\longrightarrow D be any functor that sends weak equivalences to isomorphisms. We need to show that it factors as

𝒞 F D γ ρ F˜ Ho(𝒞) \array{ \mathcal{C} && \overset{F}{\longrightarrow} && D \\ & {}_{\mathllap{\gamma}}\searrow &\Downarrow^{\rho}& \nearrow_{\mathrlap{\tilde F}} \\ && Ho(\mathcal{C}) }

uniquely up to unique natural isomorphism. Now by construction of PP and QQ in def. , γ P,Q\gamma_{P,Q} is the identity on the full subcategory of fibrant-cofibrant objects. It follows that if F˜\tilde F exists at all, it must satisfy for all XfYX \stackrel{f}{\to} Y with XX and YY both fibrant and cofibrant that

F˜([f])F(f), \tilde F([f]) \simeq F(f) \,,

(hence in particular F˜(γ P,Q(f))=F(PQf)\tilde F(\gamma_{P,Q}(f)) = F(P Q f)).

But by def. that already fixes F˜\tilde F on all of Ho(𝒞)Ho(\mathcal{C}), up to unique natural isomorphism. Hence it only remains to check that with this definition of F˜\tilde F there exists any natural isomorphism ρ\rho filling the diagram above.

To that end, apply FF to the above commuting diagram to obtain

F(X) isoF(p X) F(QX) isoF(j QX) F(PQX) F(f) F(Qf) F(PQf) F(Y) F(p y)iso F(QY) F(j QY)iso F(PQY). \array{ F(X) &\underoverset{iso}{F(p_X)}{\longleftarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) \\ {}^{\mathllap{F(f)}}\downarrow && \downarrow^{\mathrlap{F(Q f)}} && \downarrow^{\mathrlap{F(P Q f)}} \\ F(Y) &\underoverset{F(p_y)}{iso}{\longleftarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) } \,.

Here now all horizontal morphisms are isomorphisms, by assumption on FF. It follows that defining ρ XF(j QX)F(p X) 1\rho_X \coloneqq F(j_{Q X}) \circ F(p_X)^{-1} makes the required natural isomorphism:

ρ X: F(X) isoF(p X) 1 F(QX) isoF(j QX) F(PQX) = F˜(γ P,Q(X)) F(f) F(PQf) F˜(γ P,Q(f)) ρ Y: F(Y) F(p y) 1iso F(QY) F(j QY)iso F(PQY) = F˜(γ P,Q(X)). \array{ \rho_X \colon & F(X) &\underoverset{iso}{F(p_X)^{-1}}{\longrightarrow}& F(Q X) &\underoverset{iso}{F(j_{Q X})}{\longrightarrow}& F(P Q X) &=& \tilde F(\gamma_{P,Q}(X)) \\ & {}^{\mathllap{F(f)}}\downarrow && && \downarrow^{\mathrlap{F(P Q f)}} && \downarrow^{\tilde F(\gamma_{P,Q}(f))} \\ \rho_Y\colon& F(Y) &\underoverset{F(p_y)^{-1}}{iso}{\longrightarrow}& F(Q Y) &\underoverset{F(j_{Q Y})}{iso}{\longrightarrow}& F(P Q Y) &=& \tilde F(\gamma_{P,Q}(X)) } \,.
Remark

Due to theorem we may suppress the choices of cofibrant QQ and fibrant replacement PP in def. and just speak of the localization functor

γ:𝒞Ho(𝒞) \gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C})

up to natural isomorphism.

In general, the localization 𝒞[W 1]\mathcal{C}[W^{-1}] of a category with weak equivalences (𝒞,W)(\mathcal{C},W) (def. ) may invert more morphisms than just those in WW. However, if the category admits the structure of a model category (𝒞,W,Cof,Fib)(\mathcal{C},W,Cof,Fib), then its localization precisely only inverts the weak equivalences:

Proposition

(localization of model categories inverts precisely the weak equivalences)

Let 𝒞\mathcal{C} be a model category (def. ) and let γ:𝒞Ho(𝒞)\gamma \;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) be its localization functor (def. , theorem ). Then a morphism ff in 𝒞\mathcal{C} is a weak equivalence precisely if γ(f)\gamma(f) is an isomorphism in Ho(𝒞)Ho(\mathcal{C}).

(e.g. Goerss-Jardine 96, II, prop 1.14)

While the construction of the homotopy category in def. combines the restriction to good (fibrant/cofibrant) objects with the passage to homotopy classes of morphisms, it is often useful to consider intermediate stages:

Definition

Given a model category 𝒞\mathcal{C}, write

𝒞 fc 𝒞 c 𝒞 f 𝒞 \array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} }

for the system of full subcategory inclusions of:

  1. the category of fibrant objects 𝒞 f\mathcal{C}_f

  2. the category of cofibrant objects 𝒞 c\mathcal{C}_c,

  3. the category of fibrant-cofibrant objects 𝒞 fc\mathcal{C}_{fc},

all regarded a categories with weak equivalences (def. ), via the weak equivalences inherited from 𝒞\mathcal{C}, which we write (𝒞 f,W f)(\mathcal{C}_f, W_f), (𝒞 c,W c)(\mathcal{C}_c, W_c) and (𝒞 fc,W fc)(\mathcal{C}_{f c}, W_{f c}).

Remark

(categories of fibrant objects and cofibration categories)

Of course the subcategories in def. inherit more structure than just that of categories with weak equivalences from 𝒞\mathcal{C}. 𝒞 f\mathcal{C}_f and 𝒞 c\mathcal{C}_c each inherit “half” of the factorization axioms. One says that 𝒞 f\mathcal{C}_f has the structure of a “fibration category” called a “Brown-category of fibrant objects”, while 𝒞 c\mathcal{C}_c has the structure of a “cofibration category”.

We discuss properties of these categories of (co-)fibrant objects below in Homotopy fiber sequences.

The proof of theorem immediately implies the following:

Corollary

For 𝒞\mathcal{C} a model category, the restriction of the localization functor γ:𝒞Ho(𝒞)\gamma\;\colon\; \mathcal{C} \longrightarrow Ho(\mathcal{C}) from def. (using remark ) to any of the sub-categories with weak equivalences of def.

𝒞 fc 𝒞 c 𝒞 f 𝒞 γ Ho(𝒞) \array{ && \mathcal{C}_{f c} \\ & \swarrow && \searrow \\ \mathcal{C}_c && && \mathcal{C}_f \\ & \searrow && \swarrow \\ && \mathcal{C} \\ && \downarrow^{\mathrlap{\gamma}} \\ && Ho(\mathcal{C}) }

exhibits Ho(𝒞)Ho(\mathcal{C}) equivalently as the localization also of these subcategories with weak equivalences, at their weak equivalences. In particular there are equivalences of categories

Ho(𝒞)𝒞[W 1]𝒞 f[W f 1]𝒞 c[W c 1]𝒞 fc[W fc 1]. Ho(\mathcal{C}) \simeq \mathcal{C}[W^{-1}] \simeq \mathcal{C}_f[W_f^{-1}] \simeq \mathcal{C}_c[W_c^{-1}] \simeq \mathcal{C}_{f c}[W_{f c}^{-1}] \,.

The following says that for computing the hom-sets in the homotopy category, even a mixed variant of the above will do; it is sufficient that the domain is cofibrant and the codomain is fibrant:

Lemma

(hom-sets of homotopy category via mapping cofibrant resolutions into fibrant resolutions)

For X,Y𝒞X, Y \in \mathcal{C} with XX cofibrant and YY fibrant, and for P,QP, Q fibrant/cofibrant replacement functors as in def. , then the morphism

Hom Ho(𝒞)(PX,QY)=Hom 𝒞(PX,QY)/ Hom 𝒞(j X,p Y)Hom 𝒞(X,Y)/ Hom_{Ho(\mathcal{C})}(P X,Q Y) = Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, p_Y)}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim}

(on homotopy classes of morphisms, well defined by prop. ) is a natural bijection.

(Quillen 67, I.1 lemma 7)

Proof

We may factor the morphism in question as the composite

Hom 𝒞(PX,QY)/ Hom 𝒞(id PX,p Y)/ Hom 𝒞(PX,Y)/ Hom 𝒞(j X,id Y)/ Hom 𝒞(X,Y)/ . Hom_{\mathcal{C}}(P X, Q Y)/_{\sim} \overset{Hom_{\mathcal{C}}(id_{P X}, p_Y)/_\sim }{\longrightarrow} Hom_{\mathcal{C}}(P X, Y)/_{\sim} \overset{Hom_{\mathcal{C}}(j_X, id_Y)/_\sim}{\longrightarrow} Hom_{\mathcal{C}}(X,Y)/_{\sim} \,.

This shows that it is sufficient to see that for XX cofibrant and YY fibrant, then

Hom 𝒞(id X,p Y)/ :Hom 𝒞(X,QY)/ Hom 𝒞(X,Y)/ Hom_{\mathcal{C}}(id_X, p_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(X, Q Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim

is an isomorphism, and dually that

Hom 𝒞(j X,id Y)/ :Hom 𝒞(PX,Y)/ Hom 𝒞(X,Y)/ Hom_{\mathcal{C}}(j_X, id_Y)/_\sim \;\colon\; Hom_{\mathcal{C}}(P X, Y)/_\sim \to Hom_{\mathcal{C}}(X,Y)/_\sim

is an isomorphism. We discuss this for the former; the second is formally dual:

First, that Hom 𝒞(id X,p Y)Hom_{\mathcal{C}}(id_X, p_Y) is surjective is the lifting property in

QY Cof WFib p Y X f Y, \array{ \emptyset &\longrightarrow& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ X &\overset{f}{\longrightarrow}& Y } \,,

which says that any morphism f:XYf \colon X \to Y comes from a morphism f^:XQY\hat f \colon X \to Q Y under postcomposition with QYp YYQ Y \overset{p_Y}{\to} Y.

Second, that Hom 𝒞(id X,p Y)Hom_{\mathcal{C}}(id_X, p_Y) is injective is the lifting property in

XX (f,g) QY Cof WFib p Y Cyl(X) η Y, \array{ X \sqcup X &\overset{(f,g)}{\longrightarrow}& Q Y \\ {}^{\mathllap{\in Cof}}\downarrow && \downarrow^{\mathrlap{p_Y}}_{\mathrlap{\in W \cap Fib}} \\ Cyl(X) &\underset{\eta}{\longrightarrow}& Y } \,,

which says that if two morphisms f,g:XQYf, g \colon X \to Q Y become homotopic after postcomposition with p Y:QXYp_Y \colon Q X \to Y, then they were already homotopic before.

We record the following fact which will be used in part 1.1 (here):

Lemma

Let 𝒞\mathcal{C} be a model category (def. ). Then every commuting square in its homotopy category Ho(C)Ho(C) (def. ) is, up to isomorphism of squares, in the image of the localization functor 𝒞Ho(𝒞)\mathcal{C} \longrightarrow Ho(\mathcal{C}) of a commuting square in 𝒞\mathcal{C} (i.e.: not just commuting up to homotopy).

Proof

Let

A f B a b A f BHo(𝒞) \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{a}}\downarrow && \downarrow^{\mathrlap{b}} \\ A' &\underset{f'}{\longrightarrow}& B' } \;\;\;\;\; \in Ho(\mathcal{C})

be a commuting square in the homotopy category. Writing the same symbols for fibrant-cofibrant objects in 𝒞\mathcal{C} and for morphisms in 𝒞\mathcal{C} representing these, then this means that in 𝒞\mathcal{C} there is a left homotopy of the form

A f B i 1 b Cyl(A) η B i 0 f A a A. \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{b}} \\ Cyl(A) &\underset{\eta}{\longrightarrow}& B' \\ {}^{\mathllap{i_0}}\uparrow && \uparrow^{\mathrlap{f'}} \\ A &\underset{a}{\longrightarrow}& A' } \,.

Consider the factorization of the top square here through the mapping cylinder of ff

A f B i 1 (po) W Cyl(A) Cyl(f) i 0 η A B a f A \array{ A &\overset{f}{\longrightarrow}& B \\ {}^{\mathllap{i_1}}\downarrow &(po)& \downarrow^{\mathrlap{\in W}} \\ Cyl(A) &\underset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{i_0}}\uparrow &{}_{\mathllap{\eta}}\searrow& \downarrow^{\mathrlap{}} \\ A && B' \\ & {}_{\mathllap{a}}\searrow & \uparrow_{\mathrlap{f'}} \\ && A' }

This exhibits the composite Ai 0Cyl(A)Cyl(f)A \overset{i_0}{\to} Cyl(A) \to Cyl(f) as an alternative representative of ff in Ho(𝒞)Ho(\mathcal{C}), and Cyl(f)BCyl(f) \to B' as an alternative representative for bb, and the commuting square

A Cyl(f) a A f B \array{ A &\overset{}{\longrightarrow}& Cyl(f) \\ {}^{\mathllap{a}}\downarrow && \downarrow \\ A' &\underset{f'}{\longrightarrow}& B' }

as an alternative representative of the given commuting square in Ho(𝒞)Ho(\mathcal{C}).

Derived functors

Definition

(homotopical functor)

For 𝒞\mathcal{C} and 𝒟\mathcal{D} two categories with weak equivalences, def. , then a functor F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} is called a homotopical functor if it sends weak equivalences to weak equivalences.

Definition

(derived functor)

Given a homotopical functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} (def. ) between categories with weak equivalences whose homotopy categories Ho(𝒞)Ho(\mathcal{C}) and Ho(𝒟)Ho(\mathcal{D}) exist (def. ), then its (“total”) derived functor is the functor Ho(F)Ho(F) between these homotopy categories which is induced uniquely, up to unique isomorphism, by their universal property (def. ):

𝒞 F 𝒟 γ 𝒞 γ 𝒟 Ho(𝒞) Ho(F) Ho(𝒟). \array{ \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\exists \; Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,.
Remark

While many functors of interest between model categories are not homotopical in the sense of def. , many become homotopical after restriction to the full subcategories 𝒞 f\mathcal{C}_f of fibrant objects or 𝒞 c\mathcal{C}_c of cofibrant objects, def. . By corollary this is just as good for the purpose of homotopy theory.

Therefore one considers the following generalization of def. :

Definition

(left and right derived functors)

Consider a functor F:𝒞𝒟F \colon \mathcal{C} \longrightarrow \mathcal{D} out of a model category 𝒞\mathcal{C} (def. ) into a category with weak equivalences 𝒟\mathcal{D} (def. ).

  1. If the restriction of FF to the full subcategory 𝒞 f\mathcal{C}_f of fibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the right derived functor of FF and denoted by F\mathbb{R}F:

    𝒞 f 𝒞 F 𝒟 γ 𝒞 f γ 𝒟 F: 𝒞 f[W 1] Ho(𝒞) Ho(F) Ho(𝒟), \array{ & \mathcal{C}_f &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}} \downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{R} F \colon & \mathcal{C}_f[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,,

    where we use corollary .

  2. If the restriction of FF to the full subcategory 𝒞 c\mathcal{C}_c of cofibrant object becomes a homotopical functor (def. ), then the derived functor of that restriction, according to def. , is called the left derived functor of FF and denoted by 𝕃F\mathbb{L}F:

    𝒞 c 𝒞 F 𝒟 γ 𝒞 f γ 𝒟 𝕃F: 𝒞 c[W 1] Ho(𝒞) Ho(F) Ho(𝒟), \array{ & \mathcal{C}_c &\hookrightarrow& \mathcal{C} &\overset{F}{\longrightarrow}& \mathcal{D} \\ & {}^{\mathllap{\gamma}_{\mathcal{C}_f}}\downarrow && \swArrow_{\simeq} && \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ \mathbb{L} F \colon & \mathcal{C}_c[W^{-1}] &\simeq& Ho(\mathcal{C}) &\underset{Ho(F)}{\longrightarrow}& Ho(\mathcal{D}) } \,,

    where again we use corollary .

The key fact that makes def. practically relevant is the following:

Proposition

(Ken Brown's lemma)

Let 𝒞\mathcal{C} be a model category with full subcategories 𝒞 f,𝒞 c\mathcal{C}_f, \mathcal{C}_c of fibrant objects and of cofibrant objects respectively (def. ). Let 𝒟\mathcal{D} be a category with weak equivalences.

  1. A functor out of the category of fibrant objects

    F:𝒞 f𝒟 F \;\colon\; \mathcal{C}_f \longrightarrow \mathcal{D}

    is a homotopical functor, def. , already if it sends acyclic fibrations to weak equivalences.

  2. A functor out of the category of cofibrant objects

    F:𝒞 c𝒟 F \;\colon\; \mathcal{C}_c \longrightarrow \mathcal{D}

    is a homotopical functor, def. , already if it sends acyclic cofibrations to weak equivalences.

The following proof refers to the factorization lemma, whose full statement and proof we postpone to further below (lemma ).

Proof

We discuss the case of a functor on a category of fibrant objects 𝒞 f\mathcal{C}_f, def. . The other case is formally dual.

Let f:XYf \colon X \longrightarrow Y be a weak equivalence in 𝒞 f\mathcal{C}_f. Choose a path space object Path(X)Path(X) (def. ) and consider the diagram

Path(f) WFib X W p 1 *f (pb) W f Path(Y) WFibp 1 Y WFib p 0 Y, \array{ Path(f) &\underset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}_{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ Path(Y) &\overset{p_1}{\underset{\in W \cap Fib}{\longrightarrow}}& Y \\ {}^{\mathllap{p_0}}_{\mathllap{\in W \cap Fib}}\downarrow \\ Y } \,,

where the square is a pullback and Path(f)Path(f) on the top left is our notation for the universal cone object. (Below we discuss this in more detail, it is the mapping cocone of ff, def. ).

Here:

  1. p ip_i are both acyclic fibrations, by lemma ;

  2. Path(f)XPath(f) \to X is an acyclic fibration because it is the pullback of p 1p_1.

  3. p 1 *fp_1^\ast f is a weak equivalence, because the factorization lemma states that the composite vertical morphism factors ff through a weak equivalence, hence if ff is a weak equivalence, then p 1 *fp_1^\ast f is by two-out-of-three (def. ).

Now apply the functor FF to this diagram and use the assumption that it sends acyclic fibrations to weak equivalences to obtain

F(Path(f)) W F(X) F(p 1 *f) F(f) F(Path(Y)) WF(p 1) F(Y) W F(p 0) Y. \array{ F(Path(f)) &\underset{\in W }{\longrightarrow}& F(X) \\ {}^{\mathllap{F(p_1^\ast f)}}_{\mathllap{}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ F(Path(Y)) &\overset{F(p_1)}{\underset{\in W }{\longrightarrow}}& F(Y) \\ {}^{\mathllap{F(p_0)}}_{\mathllap{\in W}}\downarrow \\ Y } \,.

But the factorization lemma , in addition says that the vertical composite p 0p 1 *fp_0 \circ p_1^\ast f is a fibration, hence an acyclic fibration by the above. Therefore also F(p 0p 1 *f)F(p_0 \circ p_1^\ast f) is a weak equivalence. Now the claim that also F(f)F(f) is a weak equivalence follows with applying two-out-of-three (def. ) twice.

Corollary

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be model categories and consider F:𝒞𝒟F \colon \mathcal{C}\longrightarrow \mathcal{D} a functor. Then:

  1. If FF preserves cofibrant objects and acyclic cofibrations between these, then its left derived functor (def. ) 𝕃F\mathbb{L}F exists, fitting into a diagram

    𝒞 c F 𝒟 c γ 𝒞 γ 𝒟 Ho(𝒞) 𝕃F Ho(𝒟) \array{ \mathcal{C}_{c} &\overset{F}{\longrightarrow}& \mathcal{D}_{c} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\overset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) }
  2. If FF preserves fibrant objects and acyclic fibrants between these, then its right derived functor (def. ) F\mathbb{R}F exists, fitting into a diagram

    𝒞 f F 𝒟 f γ 𝒞 γ 𝒟 Ho(𝒞) F Ho(𝒟). \array{ \mathcal{C}_{f} &\overset{F}{\longrightarrow}& \mathcal{D}_{f} \\ {}^{\mathllap{\gamma_{\mathcal{C}}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) } \,.
Proposition

(construction of left/right derived functors)

Let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a functor between two model categories (def. ).

  1. If FF preserves fibrant objects and weak equivalences between fibrant objects, then the total right derived functor F(γ 𝒟F)\mathbb{R}F \coloneqq \mathbb{R}(\gamma_{\mathcal{D}}\circ F) (def. ) in

    𝒞 f F 𝒟 γ 𝒞 f γ 𝒟 Ho(𝒞) F Ho(𝒟) \array{ \mathcal{C}_f &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_f}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{R}F}{\longrightarrow}& Ho(\mathcal{D}) }

    is given, up to isomorphism, on any object X𝒞γ 𝒞Ho(𝒞) X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C}) by appying FF to a fibrant replacement PXP X of XX and then forming a cofibrant replacement Q(F(PX))Q(F(P X)) of the result:

F(X)Q(F(PX)). \mathbb{R}F(X) \simeq Q(F(P X)) \,.
  1. If FF preserves cofibrant objects and weak equivalences between cofibrant objects, then the total left derived functor 𝕃F𝕃(γ 𝒟F)\mathbb{L}F \coloneqq \mathbb{L}(\gamma_{\mathcal{D}}\circ F) (def. ) in

    𝒞 c F 𝒟 γ 𝒞 c γ 𝒟 Ho(𝒞) 𝕃F Ho(𝒟) \array{ \mathcal{C}_c &\overset{F}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{\gamma_{\mathcal{C}_c}}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{\mathcal{D}}}} \\ Ho(\mathcal{C}) &\underset{\mathbb{L}F}{\longrightarrow}& Ho(\mathcal{D}) }

    is given, up to isomorphism, on any object X𝒞γ 𝒞Ho(𝒞) X\in \mathcal{C} \overset{\gamma_{\mathcal{C}}}{\longrightarrow} Ho(\mathcal{C}) by appying FF to a cofibrant replacement QXQ X of XX and then forming a fibrant replacement P(F(QX))P(F(Q X)) of the result:

𝕃F(X)P(F(QX)). \mathbb{L}F(X) \simeq P(F(Q X)) \,.
Proof

We discuss the first case, the second is formally dual. By the proof of theorem we have

F(X) γ 𝒟(F(γ 𝒞)) γ 𝒟F(Q(P(X))). \begin{aligned} \mathbb{R}F(X) & \simeq \gamma_{\mathcal{D}}(F(\gamma_{\mathcal{C}})) \\ & \simeq \gamma_{\mathcal{D}}F(Q(P(X)) ) \end{aligned} \,.

But since FF is a homotopical functor on fibrant objects, the cofibrant replacement morphism F(Q(P(X)))F(P(X))F(Q(P(X)))\to F(P(X)) is a weak equivalence in 𝒟\mathcal{D}, hence becomes an isomorphism under γ 𝒟\gamma_{\mathcal{D}}. Therefore

F(X)γ 𝒟(F(P(X))). \mathbb{R}F(X) \simeq \gamma_{\mathcal{D}}(F(P(X))) \,.

Now since FF is assumed to preserve fibrant objects, F(P(X))F(P(X)) is fibrant in 𝒟\mathcal{D}, and hence γ 𝒟\gamma_{\mathcal{D}} acts on it (only) by cofibrant replacement.

Quillen adjunctions

In practice it turns out to be useful to arrange for the assumptions in corollary to be satisfied by pairs of adjoint functors (Def. ). Recall that this is a pair of functors LL and RR going back and forth between two categories

𝒞RL𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} \mathcal{D}

such that there is a natural bijection between hom-sets with LL on the left and those with RR on the right (10):

ϕ d,c:Hom 𝒞(L(d),c)Hom 𝒟(d,R(c)) \phi_{d,c} \;\colon\; Hom_{\mathcal{C}}(L(d),c) \underoverset{\simeq}{}{\longrightarrow} Hom_{\mathcal{D}}(d, R(c))

for all objects d𝒟d\in \mathcal{D} and c𝒞c \in \mathcal{C}. This being natural (Def. ) means that ϕ:Hom 𝒟(L(),)Hom 𝒞(,R())\phi \colon Hom_{\mathcal{D}}(L(-),-) \Rightarrow Hom_{\mathcal{C}}(-, R(-)) is a natural transformation, hence that for all morphisms g:d 2d 1g \colon d_2 \to d_1 and f:c 1c 2f \colon c_1 \to c_2 the following is a commuting square:

Hom 𝒞(L(d 1),c 1) ϕ d 1,c 1 Hom 𝒟(d 1,R(c 1)) L(f)()g g()R(g) Hom 𝒞(L(d 2),c 2) ϕ d 2,c 2 Hom 𝒟(d 2,R(c 2))./ \array{ Hom_{\mathcal{C}}(L(d_1), c_1) & \underoverset{\simeq}{\phi_{d_1,c_1}}{\longrightarrow} & Hom_{\mathcal{D}}(d_1, R(c_1)) \\ {}^{\mathllap{L(f) \circ (-)\circ g}}\downarrow && \downarrow^{\mathrlap{g\circ (-)\circ R(g)}} \\ Hom_{\mathcal{C}}(L(d_2), c_2) & \underoverset{\phi_{d_2, c_2}}{\simeq}{\longrightarrow} & Hom_{\mathcal{D}}(d_2, R(c_2)) } \,. /

We write (LR)(L \dashv R) to indicate such an adjunction and call LL the left adjoint and RR the right adjoint of the adjoint pair.

The archetypical example of a pair of adjoint functors is that consisting of forming Cartesian products Y×()Y \times (-) and forming mapping spaces () Y(-)^Y, as in the category of compactly generated topological spaces of def. .

If f:L(d)cf \colon L(d) \to c is any morphism, then the image ϕ d,c(f):dR(c)\phi_{d,c}(f) \colon d \to R(c) is called its adjunct, and conversely. The fact that adjuncts are in bijection is also expressed by the notation

L(c)fdcf˜R(d). \frac{ L(c) \overset{f}{\longrightarrow} d }{ c \overset{\tilde f}{\longrightarrow} R(d) } \,.

For an object d𝒟d\in \mathcal{D}, the adjunct of the identity on LdL d is called the adjunction unit η d:dRLd\eta_d \;\colon\; d \longrightarrow R L d.

For an object c𝒞c \in \mathcal{C}, the adjunct of the identity on RcR c is called the adjunction counit ϵ c:LRcc\epsilon_c \;\colon\; L R c \longrightarrow c.

Adjunction units and counits turn out to encode the adjuncts of all other morphisms by the formulas

  • (Ldfc)˜=(dηRLdRfRc)\widetilde{(L d\overset{f}{\to}c)} = (d\overset{\eta}{\to} R L d \overset{R f}{\to} R c)

  • (dgRc)˜=(LdLgLRcϵc)\widetilde{(d\overset{g}{\to} R c)} = (L d \overset{L g}{\to} L R c \overset{\epsilon}{\to} c).

Definition

(Quillen adjunction)

Let 𝒞,𝒟\mathcal{C}, \mathcal{D} be model categories. A pair of adjoint functors (Def. ) between them

(LR):𝒞RL𝒟 (L \dashv R) \;\colon\; \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {} \mathcal{D}

is called a Quillen adjunction, to be denoted

𝒞 Qu QuRL𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

and LL, RR are called left/right Quillen functors, respectively, if the following equivalent conditions are satisfied:

  1. LL preserves cofibrations and RR preserves fibrations;

  2. LL preserves acyclic cofibrations and RR preserves acyclic fibrations;

  3. LL preserves cofibrations and acyclic cofibrations;

  4. RR preserves fibrations and acyclic fibrations.

Proposition

The conditions in def. are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

First observe that

We discuss statement (i), statement (ii) is formally dual. So let f:ABf\colon A \to B be an acyclic cofibration in 𝒟\mathcal{D} and g:XYg \colon X \to Y a fibration in 𝒞\mathcal{C}. Then for every commuting diagram as on the left of the following, its (LR)(L\dashv R)-adjunct is a commuting diagram as on the right here:

A R(X) f R(g) B R(Y),L(A) X L(f) g L(B) Y. \array{ A &\longrightarrow& R(X) \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{R(g)}} \\ B &\longrightarrow& R(Y) } \;\;\;\;\;\; \,, \;\;\;\;\;\; \array{ L(A) &\longrightarrow& X \\ {}^{\mathllap{L(f)}}\downarrow && \downarrow^{\mathrlap{g}} \\ L(B) &\longrightarrow& Y } \,.

If LL preserves acyclic cofibrations, then the diagram on the right has a lift, and so the (LR)(L\dashv R)-adjunct of that lift is a lift of the left diagram. This shows that R(g)R(g) has the right lifting property against all acylic cofibrations and hence is a fibration. Conversely, if RR preserves fibrations, the same argument run from right to left gives that LL preserves acyclic fibrations.

Now by repeatedly applying (i) and (ii), all four conditions in question are seen to be equivalent.

The following is the analog of adjunction unit and adjunction counit (Def. ):

Definition

(derived adjunction unit)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be model categories (Def. ), and let

𝒞 QuRAALAA𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}

be a Quillen adjunction (Def. ). Then

  1. a derived adjunction unit at an object d𝒟d \in \mathcal{D} is a composition of the form

    Q(d)η Q(d)R(L(Q(d)))R(j L(Q(d)))R(P(L(Q(d))) Q(d) \overset{\eta_{Q(d)}}{\longrightarrow} R(L(Q(d))) \overset{R( j_{L(Q(d))} )}{\longrightarrow} R(P(L(Q(d)))

    where

    1. η\eta is the ordinary adjunction unit (Def. );

    2. Cof 𝒟i Q(d)Q(d)W 𝒟Fib 𝒟p Q(d)d\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{Q(d)}}{\longrightarrow} Q(d) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{Q(d)}}{\longrightarrow} d is a cofibrant resolution in 𝒟\mathcal{D} (Def. );

    3. L(Q(d))W 𝒞Cof 𝒞j L(Q(d))P(L(Q(d)))Fib 𝒞q L(Q(d))*L(Q(d)) \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_{L(Q(d))}}{\longrightarrow} P(L(Q(d))) \underoverset{\in Fib_{\mathcal{C}}}{q_{L(Q(d))}}{\longrightarrow} \ast is a fibrant resolution in 𝒞\mathcal{C} (Def. );

  2. a derived adjunction counit at an object c𝒞c \in \mathcal{C} is a composition of the form

    L(Q(R(P(c))))p R(P(c))LR(P(c))ϵ P(c)P(c) L(Q(R(P(c)))) \overset{ p_{R(P(c))} }{\longrightarrow} L R(P(c)) \overset{\epsilon_{P(c)}}{\longrightarrow} P(c)

    where

    1. ϵ\epsilon is the ordinary adjunction counit (Def. );

    2. cW 𝒞Cof 𝒞j cPcFib 𝒞q c*c \underoverset{\in W_{\mathcal{C}} \cap Cof_{\mathcal{C}}}{j_c}{\longrightarrow} P c \underoverset{\in Fib_{\mathcal{C}}}{q_c}{\longrightarrow} \ast is a fibrant resolution in 𝒞\mathcal{C} (Def. );

    3. Cof 𝒟i R(P(c))Q(R(P(c)))W 𝒟Fib 𝒟p R(P(c))R(P(c))\emptyset \underoverset{\in Cof_{\mathcal{D}}}{i_{R(P(c))}}{\longrightarrow} Q(R(P(c))) \underoverset{\in W_{\mathcal{D}} \cap Fib_{\mathcal{D}}}{p_{R(P(c))}}{\longrightarrow} R(P(c)) is a cofibrant resolution in 𝒟\mathcal{D} (Def. ).

We will see that Quillen adjunctions induce ordinary adjoint pairs of derived functors on homotopy categories (Prop. ). For this we first consider the following technical observation:

Lemma

(right Quillen functors preserve path space objects)

Let 𝒞RL𝒟\mathcal{C} \stackrel{\overset{L}{\longleftarrow}}{\underoverset{R}{\bot}{\longrightarrow}} \mathcal{D} be a Quillen adjunction, def. .

  1. For X𝒞X \in \mathcal{C} a fibrant object and Path(X)Path(X) a path space object (def. ), then R(Path(X))R(Path(X)) is a path space object for R(X)R(X).

  2. For X𝒞X \in \mathcal{C} a cofibrant object and Cyl(X)Cyl(X) a cylinder object (def. ), then L(Cyl(X))L(Cyl(X)) is a cylinder object for L(X)L(X).

Proof

Consider the second case, the first is formally dual.

First Observe that L(YY)LYLYL(Y \sqcup Y) \simeq L Y \sqcup L Y because LL is left adjoint and hence preserves colimits, hence in particular coproducts.

Hence

L(XXCofCyl(X))=(L(X)L(X)CofL(Cyl(X))) L(\X \sqcup X \overset{\in Cof}{\to} Cyl(X)) = (L(X) \sqcup L(X) \overset{\in Cof}{\to } L (Cyl(X)))

is a cofibration.

Second, with YY cofibrant then also YCyl(Y)Y \sqcup Cyl(Y) is a cofibrantion, since YYYY \to Y \sqcup Y is a cofibration (lemma ). Therefore by Ken Brown's lemma (prop. ) LL preserves the weak equivalence Cyl(Y)WYCyl(Y) \overset{\in W}{\longrightarrow} Y.

Proposition

(derived adjunction)

For 𝒞 Qu QuRL𝒟\mathcal{C} \underoverset{\underset{R}{\longrightarrow}}{\overset{L}{\longleftarrow}}{{}_{\phantom{Qu}}\bot_{Qu}}\mathcal{D} a Quillen adjunction, def. , also the corresponding left and right derived functors (Def. , via cor. ) form a pair of adjoint functors

Ho(𝒞)R𝕃LHo(𝒟). Ho(\mathcal{C}) \underoverset {\underset{\mathbb{R}R}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot} Ho(\mathcal{D}) \,.

Moreover, the adjunction unit and adjunction counit of this derived adjunction are the images of the derived adjunction unit and derived adjunction counit (Def. ) under the localization functors (Theorem ).

(Quillen 67, I.4 theorem 3)

Proof

For the first statement, by def. and lemma it is sufficient to see that for X,Y𝒞X, Y \in \mathcal{C} with XX cofibrant and YY fibrant, then there is a natural bijection

(93)Hom 𝒞(LX,Y)/ Hom 𝒞(X,RY)/ . Hom_{\mathcal{C}}(L X , Y)/_\sim \simeq Hom_{\mathcal{C}}(X, R Y)/_\sim \,.

Since by the adjunction isomorphism for (LR)(L \dashv R) such a natural bijection exists before passing to homotopy classes ()/ (-)/_\sim, it is sufficient to see that this respects homotopy classes. To that end, use from lemma that with Cyl(Y)Cyl(Y) a cylinder object for YY, def. , then L(Cyl(Y))L(Cyl(Y)) is a cylinder object for L(Y)L(Y). This implies that left homotopies

(f Lg):LXY (f \Rightarrow_L g) \;\colon\; L X \longrightarrow Y

given by

η:Cyl(LX)=LCyl(X)Y \eta \;\colon\; Cyl(L X) = L Cyl(X) \longrightarrow Y

are in bijection to left homotopies

(f˜ Lg˜):XRY (\tilde f \Rightarrow_L \tilde g) \;\colon\; X \longrightarrow R Y

given by

η˜:Cyl(X)RX. \tilde \eta \;\colon\; Cyl(X) \longrightarrow R X \,.

This establishes the adjunction. Now regarding the (co-)units: We show this for the adjunction unit, the case of the adjunction counit is formally dual.

First observe that for d𝒟 cd \in \mathcal{D}_c, then the defining commuting square for the left derived functor from def.

𝒟 c L 𝒞 γ P γ P,Q Ho(𝒟) 𝕃L Ho(𝒞) \array{ \mathcal{D}_c &\overset{L}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_P}}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\gamma_{P,Q}}} \\ Ho(\mathcal{D}) &\underset{\mathbb{L}L}{\longrightarrow}& Ho(\mathcal{C}) }

(using fibrant and fibrant/cofibrant replacement functors γ P\gamma_P, γ P,Q\gamma_{P,Q} from def. with their universal property from theorem , corollary ) gives that

(𝕃L)dPLPdPLdHo(𝒞), (\mathbb{L} L ) d \simeq P L P d \simeq P L d \;\;\;\; \in Ho(\mathcal{C}) \,,

where the second isomorphism holds because the left Quillen functor LL sends the acyclic cofibration j d:dPdj_d \colon d \to P d to a weak equivalence.

The adjunction unit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) on PdHo(𝒞)P d \in Ho(\mathcal{C}) is the image of the identity under

Hom Ho(𝒞)((𝕃L)Pd,(𝕃L)Pd)Hom Ho(𝒞)(Pd,(R)(𝕃L)Pd). Hom_{Ho(\mathcal{C})}((\mathbb{L}L) P d, (\mathbb{L} L) P d) \overset{\simeq}{\to} Hom_{Ho(\mathcal{C})}(P d, (\mathbb{R}R)(\mathbb{L}L) P d) \,.

By the above and the proof of prop. , that adjunction isomorphism is equivalently that of (LR)(L \dashv R) under the isomorphism

Hom Ho(𝒞)(PLd,PLd)Hom(j Ld,id)Hom 𝒞(Ld,PLd)/ Hom_{Ho(\mathcal{C})}(P L d , P L d) \overset{Hom(j_{L d}, id)}{\longrightarrow} Hom_{\mathcal{C}}(L d, P L d)/_\sim

of lemma . Hence the derived adjunction unit (Def. ) is the (LR)(L \dashv R)-adjunct of

Ldj LdPLdidPLd, L d \overset{j_{L d}}{\longrightarrow} P L d \overset{id}{\to} P L d \,,

which indeed (by the formula for adjuncts, Prop. ) is the derived adjunction unit

XηRLdR(j Ld)RPLd. X \overset{\eta}{\longrightarrow} R L d \overset{R (j_{L d})}{\longrightarrow} R P L d \,.

This suggests to regard passage to homotopy categories and derived functors as itself being a suitable functor from a category of model categories to the category of categories. Due to the role played by the distinction between left Quillen functors and right Quillen functors, this is usefully formulated as a double functor:

Definition

(double category of model categories)

The (very large) double category of model categories ModCat dblModCat_{dbl} is the double category (Def. ) that has

  1. as objects: model categories 𝒞\mathcal{C} (Def. );

  2. as vertical morphisms: left Quillen functors 𝒞L\mathcal{C} \overset{L}{\longrightarrow} \mathcal{E} (Def. );

  3. as horizontal morphisms: right Quillen functors 𝒞R𝒟\mathcal{C} \overset{R}{\longrightarrow}\mathcal{D} (Def. );

  4. as 2-morphisms natural transformations between the composites of underlying functors:

    L 2R 1ϕR 2L 1AAAAA𝒞 AAR 1AA 𝒟 L 1 ϕ L 2 𝒞 AAR 2AA 𝒟 L_2\circ R_1 \overset{\phi}{\Rightarrow} R_2\circ L_1 \phantom{AAAAA} \array{ \mathcal{C} &\overset{\phantom{AA}R_1\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{C} &\underset{\phantom{AA}R_2\phantom{AA}}{\longrightarrow}& \mathcal{D} }

and composition is given by ordinary composition of functors, horizontally and vertically, and by whiskering-composition of natural transformations.

(Shulman 07, Example 4.6)

There is hence a forgetful double functor (Remark )

F:ModCat dblSq(Cat) F \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)

to the double category of squares (Example ) in the 2-category of categories (Example ), which forgets the model category-structure and the Quillen functor-property.

The following records the 2-functoriality of sending Quillen adjunctions to adjoint pairs of derived functors (Prop. ):

Proposition

(homotopy double pseudofunctor on the double category of model categories)

There is a double pseudofunctor (Remark )

Ho():ModCat dblSq(Cat) Ho(-) \;\colon\; ModCat_{dbl} \longrightarrow Sq(Cat)

from the double category of model categories (Def. ) to the double category of squares (Example ) in the 2-category Cat (Example ), which sends

  1. a model category 𝒞\mathcal{C} to its homotopy category of a model category (Def. );

  2. a left Quillen functor (Def. ) to its left derived functor (Def. );

  3. a right Quillen functor (Def. ) to its right derived functor (Def. );

  4. a natural transformation

    𝒞 R 1 𝒟 L 1 ϕ L 2 R 2 \array{ \mathcal{C} &\overset{R_1}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{L_1}}\Big\downarrow &{}^{\mathllap{ \phi }}\swArrow& \Big\downarrow{}^{\mathrlap{L_2}} \\ \mathcal{E} &\underset{R_2}{\longrightarrow}& \mathcal{F} }

    to the “derived natural transformation

    Ho(𝒞) R 1 Ho(𝒟) 𝕃L 1 Ho(ϕ) 𝕃L 2 Ho() R 2 Ho() \array{ Ho(\mathcal{C}) &\overset{\mathbb{R}R_1}{\longrightarrow}& Ho(\mathcal{D}) \\ {}^{\mathllap{\mathbb{L}L_1}}\Big\downarrow &\overset{Ho(\phi)}{\swArrow}& \Big\downarrow{}^{\mathrlap{\mathbb{L}L_2}} \\ Ho(\mathcal{E}) &\underset{\mathbb{R}R_2}{\longrightarrow}& Ho(\mathcal{F}) }

    given by the zig-zag

    (94)Ho(ϕ):L 2QR 1PL 2QR 1QPL 2R 1QPϕR 2L 1QPR 2PL1QPR 2RL 1Q, Ho(\phi) \;\colon\; L_2 Q R_1 P \overset{}{\longleftarrow} L_2 Q R_1 Q P \longrightarrow L_2 R_1 Q P \overset{\phi}{\longrightarrow} R_2 L_1 Q P \longrightarrow R_2 P L1 Q P \longleftarrow R_2 R L_1 Q \,,

    where the unlabeled morphisms are induced by fibrant resolution cPcc \to P c and cofibrant resolution QccQ c \to c, respectively (Def. ).

(Shulman 07, Theorem 7.6)

Lemma

(recognizing derived natural isomorphisms)

For the derived natural transformation Ho(ϕ)Ho(\phi) in (94) to be invertible in the homotopy category, it is sufficient that for every object c𝒞c \in \mathcal{C} which is both fibrant and cofibrant the following composite natural transformation

R 2QL 1cR 2p L 1cR 2L 1cϕL 2R 1cL 2j R 1cL 2PR 1c R_2 Q L_1 c \overset{ R_2 p_{L_1 c} }{\longrightarrow} R_2 L_1 c \overset{\phi}{\longrightarrow} L_2 R_1 c \overset{ L_2 j_{R_1 c} }{\longrightarrow} L_2 P R_1 c

(of ϕ\phi with images of fibrant resolution/cofibrant resolution, Def. ) is invertible in the homotopy category, hence that the composite is a weak equivalence (by Prop. ).

(Shulman 07, Remark 7.2)

Example

(derived functor of left-right Quillen functor)

Let 𝒞\mathcal{C}, 𝒟\mathcal{D} be model categories (Def. ), and let

𝒞AFA𝒞 \mathcal{C} \overset{\phantom{A}F\phantom{A}}{\longrightarrow} \mathcal{C}

be a functor that is both a left Quillen functor as well as a right Quillen functor (Def. ). This means equivalently that there is a 2-morphism in the double category of model categories (Def. ) of the form

(95)𝒞 AAFAA 𝒟 F id id 𝒟 AidA 𝒟 \array{ \mathcal{C} &\overset{\phantom{AA}F\phantom{AA}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{F}}\Big\downarrow &{}^{id}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{D} &\underset{\phantom{A}id\phantom{A}}{\longrightarrow}& \mathcal{D} }

It follows that the left derived functor 𝕃F\mathbb{L}F and right derived functor F\mathbb{R}F of FF (Def. ) are naturally isomorphic:

Ho(𝒞)𝕃FFHo(𝒟). Ho(\mathcal{C}) \overset{ \mathbb{L}F \simeq \mathbb{R}F }{\longrightarrow} Ho(\mathcal{D}) \,.

(Shulman 07, corollary 7.8)

Proof

To see the natural isomorphism 𝕃FF\mathbb{L}F \simeq \mathbb{R}F: By Prop. this is implied once the derived natural transformation Ho(id)Ho(id) of (95) is a natural isomorphism. By Prop. this is the case, in the present situation, if the composition of

QFcp FcFcj FcPFc Q F c \overset{ p_{F c} }{\longrightarrow} F c \overset{ j_{F c} }{\longrightarrow} P F c

is a weak equivalence. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution (Def. ).

The following is the analog of co-reflective subcategories (Def. ) for model categories:

Definition

(Quillen reflection)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be model categories (Def. ), and let

𝒞 QuAARAAL𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}

be a Quillen adjunction between them (Def. ). Then this may be called

  1. a Quillen reflection if the derived adjunction counit (Def. ) is componentwise a weak equivalence;

  2. a Quillen co-reflection if the derived adjunction unit (Def. ) is componentwise a weak equivalence.

The main class of examples of Quillen reflections are left Bousfield localizations, discussed as Prop. below.

Proposition

(characterization of Quillen reflections)

Let

𝒞 QuAARAAL𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot_{Qu}} \mathcal{D}

be a Quillen adjunction (Def. ) and write

Ho(𝒞) QuAARAA𝕃LHo(𝒟) Ho(\mathcal{C}) \underoverset {\underset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow}} {\overset{\mathbb{L}L}{\longleftarrow}} {\bot_{Qu}} Ho(\mathcal{D})

for the induced adjoint pair of derived functors on the homotopy categories, from Prop. .

Then

  1. (LQuR)(L \underset{Qu}{\dashv} R) is a Quillen reflection (Def. ) precisely if (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) is a reflective subcategory-inclusion (Def. );

  2. (LQuR)(L \underset{Qu}{\dashv} R) is a Quillen co-reflection] (Def. ) precisely if (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) is a co-reflective subcategory-inclusion (Def. );

Proof

By Prop. the components of the adjunction unit/counit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) are precisely the images under localization of the derived adjunction unit/counit of (LQuR)(L \underset{Qu}{\dashv} R). Moreover, by Prop. the localization functor of a model category inverts precisely the weak equivalences. Hence the adjunction (co-)unit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) is an isomorphism if and only if the derived (co-)unit of (LQuR)(L \underset{Qu}{\dashv} R) is a weak equivalence, respectively.

With this the statement reduces to the characterization of (co-)reflections via invertible units/counits, respectively, from Prop. .

The following is the analog of adjoint equivalence of categories (Def. ) for model categories:

Definition

(Quillen equivalence)

For 𝒞,𝒟\mathcal{C}, \mathcal{D} two model categories (Def. ), a Quillen adjunction (def. )

𝒞 Qu QuAARAAL𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

is called a Quillen equivalence, to be denoted

𝒞 Qu QuRAALAA𝒟, \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{D} \,,

if the following equivalent conditions hold:

  1. The right derived functor of RR (via prop. , corollary ) is an equivalence of categories

    R:Ho(𝒞)Ho(𝒟). \mathbb{R}R \colon Ho(\mathcal{C}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{D}) \,.
  2. The left derived functor of LL (via prop. , corollary ) is an equivalence of categories

    𝕃L:Ho(𝒟)Ho(𝒞). \mathbb{L}L \colon Ho(\mathcal{D}) \overset{\simeq}{\longrightarrow} Ho(\mathcal{C}) \,.
  3. For every cofibrant object d𝒟d\in \mathcal{D}, the derived adjunction unit (Def. )

    dη dR(L(d))R(j L(d))R(P(L(d))) d \overset{\eta_d}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d)))

    is a weak equivalence;

    and for every fibrant object c𝒞c \in \mathcal{C}, the derived adjunction counit (Def. )

    L(Q(R(c)))L(p R(c))L(R(c))ϵc L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c

    is a weak equivalence.

  4. For every cofibrant object d𝒟d \in \mathcal{D} and every fibrant object c𝒞c \in \mathcal{C}, a morphism dR(c)d \longrightarrow R(c) is a weak equivalence precisely if its adjunct morphism L(c)dL(c) \to d is:

    dW 𝒟R(c)L(d)W 𝒞c. \frac{ d \overset{\in W_{\mathcal{D}}}{\longrightarrow} R(c) }{ L(d) \overset{\in W_{\mathcal{C}}}{\longrightarrow} c } \,.
Poposition

The conditions in def. are indeed all equivalent.

(Quillen 67, I.4, theorem 3)

Proof

That 1)2)1) \Leftrightarrow 2) follows from prop. (if in an adjoint pair one is an equivalence, then so is the other).

To see the equivalence 1),2)3)1),2) \Leftrightarrow 3), notice (prop.) that a pair of adjoint functors is an equivalence of categories precisely if both the adjunction unit and the adjunction counit are natural isomorphisms. Hence it is sufficient to see that the derived adjunction unit/derived adjunction counit (Def. ) indeed represent the adjunction (co-)unit of (𝕃LR)(\mathbb{L}L \dashv \mathbb{R}R) in the homotopy category. But this is the statement of Prop. .

To see that 4)3)4) \Rightarrow 3):

Consider the weak equivalence LXj LXPLXL X \overset{j_{L X}}{\longrightarrow} P L X. Its (LR)(L \dashv R)-adjunct is

XηRLXRj LXRPLX X \overset{\eta}{\longrightarrow} R L X \overset{R j_{L X}}{\longrightarrow} R P L X

by assumption 4) this is again a weak equivalence, which is the requirement for the derived adjunction unit in 3). Dually for derived adjunction counit.

To see 3)4)3) \Rightarrow 4):

Consider any f:Ldcf \colon L d \to c a weak equivalence for cofibrant dd, firbant cc. Its adjunct f˜\tilde f sits in a commuting diagram

f˜: d η RLd Rf Rc = Rj Ld Rj c d W RPLd RPf RPc, \array{ \tilde f \colon & d &\overset{\eta}{\longrightarrow}& R L d &\overset{R f}{\longrightarrow}& R c \\ & {}^{\mathllap{=}}\downarrow && \downarrow^{\mathrlap{R j_{L d}}} && \downarrow^{\mathrlap{R j_c}} \\ & d &\underset{\in W}{\longrightarrow}& R P L d &\overset{R P f}{\longrightarrow}& R P c } \,,

where PfP f is any lift constructed as in def. .

This exhibits the bottom left morphism as the derived adjunction unit (Def. ), hence a weak equivalence by assumption. But since ff was a weak equivalence, so is PfP f (by two-out-of-three). Thereby also RPfR P f and Rj YR j_Y, are weak equivalences by Ken Brown's lemma and the assumed fibrancy of cc. Therefore by two-out-of-three (def. ) also the adjunct f˜\tilde f is a weak equivalence.

Example

(trivial Quillen equivalence)

Let 𝒞\mathcal{C} be a model category (Def. ). Then the identity functor on 𝒞\mathcal{C} constitutes a Quillen equivalence (Def. ) from 𝒞\mathcal{C} to itself:

𝒞 Qu Quidid𝒞 \mathcal{C} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}
Proof

From prop. it is clear that in this case the derived functors 𝕃id\mathbb{L}id and id\mathbb{R}id both are themselves the identity functor on the homotopy category of a model category, hence in particular are an equivalence of categories.

In certain situations the conditions on a Quillen equivalence simplify. For instance:

Proposition

(recognition of Quillen equivalences)

If in a Quillen adjunction 𝒞 RL 𝒟 \array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}} (def. ) the right adjoint RR “creates weak equivalences” (in that a morphism ff in 𝒞\mathcal{C} is a weak equivalence precisly if U(f)U(f) is) then (LR)(L \dashv R) is a Quillen equivalence (def. ) precisely already if for all cofibrant objects d𝒟d \in \mathcal{D} the plain adjunction unit

dηR(L(d)) d \overset{\eta}{\longrightarrow} R (L (d))

is a weak equivalence.

Proof

By prop. , generally, (LR)(L \dashv R) is a Quillen equivalence precisely if

  1. for every cofibrant object d𝒟d\in \mathcal{D}, the derived adjunction unit (Def. )

    dηR(L(d))R(j L(d))R(P(L(d))) d \overset{\eta}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d)))

    is a weak equivalence;

  2. for every fibrant object c𝒞c \in \mathcal{C}, the derived adjunction counit (Def. )

    L(Q(R(c)))L(p R(c))L(R(c))ϵc L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c

    is a weak equivalence.

Consider the first condition: Since RR preserves the weak equivalence j L(d)j_{L(d)}, then by two-out-of-three (def. ) the composite in the first item is a weak equivalence precisely if η\eta is.

Hence it is now sufficient to show that in this case the second condition above is automatic.

Since RR also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image

R(L(Q(R(c))))R(L(p R(c)))R(L(R(c)))R(ϵ)R(c) R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c)

under RR is.

Moreover, assuming, by the above, that η Q(R(c))\eta_{Q(R(c))} on the cofibrant object Q(R(c))Q(R(c)) is a weak equivalence, then by two-out-of-three this composite is a weak equivalence precisely if the further composite with η\eta is

Q(R(c))η Q(R(c))R(L(Q(R(c))))R(L(p R(c)))R(L(R(c)))R(ϵ)R(c). Q(R(c)) \overset{\eta_{Q(R(c))}}{\longrightarrow} R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) \,.

By the formula for adjuncts, this composite is the (LR)(L\dashv R)-adjunct of the original composite, which is just p R(c)p_{R(c)}

L(Q(R(c)))L(p R(c))L(R(c))ϵcQ(R(C))p R(c)R(c). \frac{ L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c }{ Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c) } \,.

But p R(c)p_{R(c)} is a weak equivalence by definition of cofibrant replacement.

The following is the analog of adjoint triples, adjoint quadruples (Remark ), etc. for model categories:

Definition

(Quillen adjoint triple)

Let 𝒞 1,𝒞 2,𝒟\mathcal{C}_1, \mathcal{C}_2, \mathcal{D} be model categories (Def. ), where 𝒞 1\mathcal{C}_1 and 𝒞 2\mathcal{C}_2 share the same underlying category 𝒞\mathcal{C}, and such that the identity functor on 𝒞\mathcal{C} constitutes a Quillen equivalence (Def. ):

𝒞 2 Qu QuAAidAAAAidAA𝒞 1 \mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C}_1

Then

  1. a Quillen adjoint triple of the form

    𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟 \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D}

    is diagrams in the double category of model categories (Def. ) of the form

    𝒞 1 AAidAA 𝒞 2 L η id 𝒞 2 ARA 𝒟 ACA 𝒞 1 id ϵ C id id 𝒞 2 AAidAA 𝒞 2 AAidAA 𝒞 2 \array{ && \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 \\ && {}^{\mathllap{ L }}\Big\downarrow &{}^{\mathllap{\eta}}\swArrow& \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\overset{ \phantom{A}R\phantom{A} }{\longrightarrow}& \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{ id }}\Big\downarrow & {}^{\mathllap{\epsilon}}\swArrow & {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }

    such that η\eta is the unit of an adjunction and ϵ\epsilon the counit of an adjunction, thus exhibiting Quillen adjunctions

    𝒞 1 Qu QuCL𝒟 𝒞 2 Qu QuRC𝒟 \array{ \mathcal{C}_1 \underoverset {\underset{C}{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }

    and such that the derived natural transformation Ho(id)Ho(id) of the bottom right square (94) is invertible (a natural isomorphism);

  2. a Quillen adjoint triple of the form

    𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟 \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longrightarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longleftarrow} \\ } \mathcal{D}

    is diagram in the double category of model categories (Def. ) of the form

    𝒞 2 AAidAA 𝒞 1 AAidAA 𝒞 1 id id C ϵ id 𝒞 2 ACA 𝒟 R 𝒞 1 id ϵ L 𝒞 2 AAidAA 𝒞 2 \array{ \mathcal{C}_2 &\overset{ \phantom{AA} id \phantom{AA} }{\longrightarrow}& \mathcal{C}_1 &\overset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}} \Big\downarrow &{}^{ \mathllap{ id } }\swArrow& \Big\downarrow{}^{ \mathrlap{ C } } & {}^{ \mathllap{\epsilon} }\swArrow & \Big\downarrow{}^{\mathrlap{id}} \\ \mathcal{C}_2 &\underset{ \phantom{A}C\phantom{A} }{\longrightarrow}& \mathcal{D} &\underset{R}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{id}}\Big\downarrow &{}^{\mathllap{ \epsilon }}\swArrow& \Big\downarrow{}^{\mathrlap{L}} \\ \mathcal{C}_2 &\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}& \mathcal{C}_2 }

    such that η\eta is the unit of an adjunction and ϵ\epsilon the counit of an adjunction, thus exhibiting Quillen adjunctions

    𝒞 2 Qu QuCL𝒟 𝒞 1 Qu QuRC𝒟 \array{ \mathcal{C}_2 \underoverset {\underset{C}{\longrightarrow}} {\overset{L}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} \\ \\ \mathcal{C}_1 \underoverset {\underset{R}{\longleftarrow}} {\overset{C}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D} }

    and such that the derived natural transformation Ho(id)Ho(id) of the top left square square (here) is invertible (a natural isomorphism).

If a Quillen adjoint triple of the first kind overlaps with one of the second kind

𝒞 1/2 Qu QuL 1=A a Qu QuC 1=L 2 Qu QuR 1=C 2 A a=R 2 𝒟 \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L_1 \phantom{= A_a}}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C_1 = L_2}{\longleftarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{R_1 = C_2}{\longrightarrow} \\ \overset{\phantom{A_a = } R_2}{\longleftarrow} \\ } \mathcal{D}

we speak of a Quillen adjoint quadruple, and so forth.

Proposition

(Quillen adjoint triple induces adjoint triple of derived functors on homotopy categories)

Given a Quillen adjoint triple (Def. ), the induced derived functors (Def. ) on the homotopy categories form an ordinary adjoint triple (Remark ):

𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟AAAAHo()AAAAHo(𝒞) Qu Qu𝕃L Qu Qu𝕃CC AARAA Ho(𝒟) \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})

\,

𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟AAAAHo()AAAAHo(𝒞) Qu Qu𝕃L Qu Qu𝕃CC AARAA Ho(𝒟) \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})
Proof

This follows immediately from the fact that passing to homotopy categories of model categories is a double pseudofunctor from the double category of model categories to the double category of squares in Cat (Prop. ).

\,

Mapping cones

In the context of homotopy theory, a pullback diagram, such as in the definition of the fiber in example

fib(f) X f * Y \array{ fib(f) &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }

ought to commute only up to a (left/right) homotopy (def. ) between the outer composite morphisms. Moreover, it should satisfy its universal property up to such homotopies.

Instead of going through the full theory of what this means, we observe that this is plausibly modeled by the following construction, and then we check (below) that this indeed has the relevant abstract homotopy theoretic properties.

Definition

Let 𝒞\mathcal{C} be a model category, def. with 𝒞 */\mathcal{C}^{\ast/} its model structure on pointed objects, prop. . For f:XYf \colon X \longrightarrow Y a morphism between cofibrant objects (hence a morphism in (𝒞 */) c𝒞 */(\mathcal{C}^{\ast/})_c\hookrightarrow \mathcal{C}^{\ast/}, def. ), its reduced mapping cone is the object

Cone(f)*XCyl(X)XY Cone(f) \coloneqq \ast \underset{X}{\sqcup} Cyl(X) \underset{X}{\sqcup} Y

in the colimiting diagram

X f Y i 1 i X i 0 Cyl(X) η * Cone(f), \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} && \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) \\ \downarrow && & \searrow^{\mathrlap{\eta}} & \downarrow \\ {*} &\longrightarrow& &\longrightarrow& Cone(f) } \,,

where Cyl(X)Cyl(X) is a cylinder object for XX, def. .

Dually, for f:XYf \colon X \longrightarrow Y a morphism between fibrant objects (hence a morphism in (𝒞 *) f𝒞 */(\mathcal{C}^{\ast})_f\hookrightarrow \mathcal{C}^{\ast/}, def. ), its mapping cocone is the object

Path *(f)*×YPath(Y)×YY Path_\ast(f) \coloneqq \ast \underset{Y}{\times} Path(Y)\underset{Y}{\times} Y

in the following limit diagram

Path *(f) X η f Path(Y) p 1 Y p 0 * Y, \array{ Path_\ast(f) &\longrightarrow& &\longrightarrow& X \\ \downarrow &\searrow^{\mathrlap{\eta}}& && \downarrow^{\mathrlap{f}} \\ && Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow && \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,,

where Path(Y)Path(Y) is a path space object for YY, def. .

Remark

When we write homotopies (def. ) as double arrows between morphisms, then the limit diagram in def. looks just like the square in the definition of fibers in example , except that it is filled by the right homotopy given by the component map denoted η\eta:

Path *(f) X η f * Y. \array{ Path_\ast(f) &\longrightarrow& X \\ \downarrow &\swArrow_{\eta}& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,.

Dually, the colimiting diagram for the mapping cone turns to look just like the square for the cofiber, except that it is filled with a left homotopy

X f Y η * Cone(f) \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &\swArrow_{\eta}& \downarrow \\ \ast &\longrightarrow& Cone(f) }
Proposition

The colimit appearing in the definition of the reduced mapping cone in def. is equivalent to three consecutive pushouts:

X f Y i 1 (po) i X i 0 Cyl(X) Cyl(f) (po) (po) * Cone(X) Cone(f). \array{ && X &\stackrel{f}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{i}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cyl(f) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& Cone(f) } \,.

The two intermediate objects appearing here are called

  • the plain reduced cone Cone(X)*XCyl(X)Cone(X) \coloneqq \ast \underset{X}{\sqcup} Cyl(X);

  • the reduced mapping cylinder Cyl(f)Cyl(X)XYCyl(f) \coloneqq Cyl(X) \underset{X}{\sqcup} Y.

Dually, the limit appearing in the definition of the mapping cocone in def. is equivalent to three consecutive pullbacks:

Path *(f) Path(f) X (pb) (pb) f Path *(Y) Path(Y) p 1 Y (pb) p 0 * Y. \array{ Path_\ast(f) &\longrightarrow& Path(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path_\ast(Y) &\longrightarrow& Path(Y) &\underset{p_1}{\longrightarrow}& Y \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& Y } \,.

The two intermediate objects appearing here are called

  • the based path space object Path *(Y)*YPath(Y)Path_\ast(Y) \coloneqq \ast \underset{Y}{\prod} Path(Y);

  • the mapping path space or mapping co-cylinder Path(f)X×YPath(X)Path(f) \coloneqq X \underset{Y}{\times} Path(X).

Definition

Let X𝒞 */X \in \mathcal{C}^{\ast/} be any pointed object.

  1. The mapping cone, def. , of X*X \to \ast is called the reduced suspension of XX, denoted

    ΣX=Cone(X*). \Sigma X = Cone(X\to\ast)\,.

    Via prop. this is equivalently the coproduct of two copies of the cone on XX over their base:

    X * i 1 (po) X i 0 Cyl(X) Cone(X) (po) (po) * Cone(X) ΣX. \array{ && X &\stackrel{}{\longrightarrow}& \ast \\ && \downarrow^{\mathrlap{i_1}} &(po)& \downarrow^{\mathrlap{}} \\ X &\stackrel{i_0}{\longrightarrow}& Cyl(X) &\longrightarrow& Cone(X) \\ \downarrow &(po)& \downarrow & (po) & \downarrow \\ {*} &\longrightarrow& Cone(X) &\longrightarrow& \Sigma X } \,.

    This is also equivalently the cofiber, example of (i 0,i 1)(i_0,i_1), hence (example ) of the wedge sum inclusion:

    XX=XX(i 0,i 1)Cyl(X)cofib(i 0,i 1)ΣX. X \vee X = X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} Cyl(X) \overset{cofib(i_0,i_1)}{\longrightarrow} \Sigma X \,.
  2. The mapping cocone, def. , of *X\ast \to X is called the loop space object of XX, denoted

    ΩX=Path *(*X). \Omega X = Path_\ast(\ast \to X) \,.

    Via prop. this is equivalently

    ΩX Path *(X) * (pb) (pb) Path *(X) Path(X) p 1 X (pb) p 0 * X. \array{ \Omega X &\longrightarrow& Path_\ast(X) &\longrightarrow& \ast \\ \downarrow &(pb)& \downarrow &(pb)& \downarrow^{} \\ Path_\ast(X) &\longrightarrow& Path(X) &\underset{p_1}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{p_0}} \\ \ast &\longrightarrow& X } \,.

    This is also equivalently the fiber, example of (p 0,p 1)(p_0,p_1):

    ΩXfib(p 0,p 1)Path(X)(p 0,p 1)X×X. \Omega X \overset{fib(p_0,p_1)}{\longrightarrow} Path(X) \overset{(p_0,p_1)}{\longrightarrow} X \times X \,.
Proposition

In pointed topological spaces Top */Top^{\ast/},

  • the reduced suspension objects (def. ) induced from the standard reduced cylinder ()(I +)(-)\wedge (I_+) of example are isomorphic to the smash product (def. ) with the 1-sphere, for later purposes we choose to smash on the left and write

    cofib(XXX(I +))S 1X, cofib(X \vee X \to X \wedge (I_+)) \simeq S^1 \wedge X \,,

Dually:

  • the loop space objects (def. ) induced from the standard pointed path space object Maps(I +,) *Maps(I_+,-)_\ast are isomorphic to the pointed mapping space (example ) with the 1-sphere

    fib(Maps(I +,X) *X×X)Maps(S 1,X) *. fib(Maps(I_+,X)_\ast \to X \times X) \simeq Maps(S^1, X)_\ast \,.
Proof

By immediate inspection: For instance the fiber of Maps(I +,X) *X×XMaps(I_+,X)_\ast \longrightarrow X\times X is clearly the subspace of the unpointed mapping space X IX^I on elements that take the endpoints of II to the basepoint of XX.

Example

For 𝒞=\mathcal{C} = Top with Cyl(X)=X×ICyl(X) = X\times I the standard cyclinder object, def. , then by example , the mapping cone, def. , of a continuous function f:XYf \colon X \longrightarrow Y is obtained by

  1. forming the cylinder over XX;

  2. attaching to one end of that cylinder the space YY as specified by the map ff.

  3. shrinking the other end of the cylinder to the point.

Accordingly the suspension of a topological space is the result of shrinking both ends of the cylinder on the object two the point. This is homeomoprhic to attaching two copies of the cone on the space at the base of the cone.

(graphics taken from Muro 2010)

Below in example we find the homotopy-theoretic interpretation of this standard topological mapping cone as a model for the homotopy cofiber.

Remark

The formula for the mapping cone in prop. (as opposed to that of the mapping co-cone) does not require the presence of the basepoint: for f:XYf \colon X \longrightarrow Y a morphism in 𝒞\mathcal{C} (as opposed to in 𝒞 */\mathcal{C}^{\ast/}) we may still define

Cone(f)YXCone(X), Cone'(f) \coloneqq Y \underset{X}{\sqcup} Cone'(X) \,,

where the prime denotes the unreduced cone, formed from a cylinder object in 𝒞\mathcal{C}.

Proposition

For f:XYf \colon X \longrightarrow Y a morphism in Top, then its unreduced mapping cone, remark , with respect to the standard cylinder object X×IX \times I def. , is isomorphic to the reduced mapping cone, def. , of the morphism f +:X +Y +f_+ \colon X_+ \to Y_+ (with a basepoint adjoined, def. ) with respect to the standard reduced cylinder (example ):

Cone(f)Cone(f +). Cone'(f) \simeq Cone(f_+) \,.
Proof

By prop. and example , Cone(f +)Cone(f_+) is given by the colimit in TopTop over the following diagram:

* X* (f,id) Y* X* (X×I)* * Cone(f +). \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone(f_+) } \,.

We may factor the vertical maps to give

* X* (f,id) Y* X* (X×I)* ** Cone(f) + * Cone(f). \array{ \ast &\longrightarrow& X \sqcup \ast &\overset{(f,id)}{\longrightarrow}& Y \sqcup \ast \\ \downarrow && \downarrow && \downarrow \\ X \sqcup\ast &\longrightarrow& (X \times I) \sqcup \ast \\ \downarrow && && \downarrow \\ \ast \sqcup \ast &\longrightarrow& &\longrightarrow& Cone'(f)_+ \\ \downarrow && && \downarrow \\ \ast &\longrightarrow& &\longrightarrow& Cone'(f) } \,.

This way the top part of the diagram (using the pasting law to compute the colimit in two stages) is manifestly a cocone under the result of applying () +(-)_+ to the diagram for the unreduced cone. Since () +(-)_+ is itself given by a colimit, it preserves colimits, and hence gives the partial colimit Cone(f) +Cone'(f)_+ as shown. The remaining pushout then contracts the remaining copy of the point away.

Example makes it clear that every cycle S nYS^n \to Y in YY that happens to be in the image of XX can be continuously translated in the cylinder-direction, keeping it constant in YY, to the other end of the cylinder, where it shrinks away to the point. This means that every homotopy group of YY, def. , in the image of ff vanishes in the mapping cone. Hence in the mapping cone the image of XX under ff in YY is removed up to homotopy. This makes it intuitively clear how Cone(f)Cone(f) is a homotopy-version of the cokernel of ff. We now discuss this formally.

Lemma

(factorization lemma)

Let 𝒞 c\mathcal{C}_c be a category of cofibrant objects, def. . Then for every morphism f:XYf \colon X \longrightarrow Y the mapping cylinder-construction in def. provides a cofibration resolution of ff, in that

  1. the composite morphism Xi 0Cyl(X)(i 1) *fCyl(f)X \overset{i_0}{\longrightarrow} Cyl(X) \overset{(i_1)_\ast f}{\longrightarrow} Cyl(f) is a cofibration;

  2. ff factors through this morphism by a weak equivalence left inverse to an acyclic cofibration

    f:XCof(i 1) *fi 0Cyl(f)WY, f \;\colon\; X \underoverset{\in Cof}{(i_1)_\ast f\circ i_0}{\longrightarrow} Cyl(f) \underset{\in W}{\longrightarrow} Y \,,

Dually:

Let 𝒞 f\mathcal{C}_f be a category of fibrant objects, def. . Then for every morphism f:XYf \colon X \longrightarrow Y the mapping cocylinder-construction in def. provides a fibration resolution of ff, in that

  1. the composite morphism Path(f)p 1 *fPath(Y)p 0YPath(f) \overset{p_1^\ast f}{\longrightarrow} Path(Y) \overset{p_0}{\longrightarrow} Y is a fibration;

  2. ff factors through this morphism by a weak equivalence right inverse to an acyclic fibration:

    f:XWPath(f)Fibp 0p 1 *fY, f \;\colon\; X \underset{\in W}{\longrightarrow} Path(f) \underoverset{\in Fib}{p_0 \circ p_1^\ast f}{\longrightarrow} Y \,,
Proof

We discuss the second case. The first case is formally dual.

So consider the mapping cocylinder-construction from prop.

Path(f) WFib X p 1 *f (pb) f Path(Y) WFibp 1 Y WFib p 0 Y. \array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{p_1^\ast f}}\downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{\in W \cap Fib}}\downarrow^{\mathrlap{p_0}} \\ Y } \,.

To see that the vertical composite is indeed a fibration, notice that, by the pasting law, the above pullback diagram may be decomposed as a pasting of two pullback diagram as follows

Path(f) Fib(f,id) *(p 1,p 0) X×Y pr 1 X (f,Id) f Path(Y) (p 1,p 0)Fib Y×Y pr 1 Y p 0 pr 2Fib Y. \array{ Path(f) &\underoverset{\in Fib}{(f,id)^\ast(p_1,p_0)}{\longrightarrow}& X \times Y &\stackrel{pr_1}{\to}& X \\ \downarrow && \downarrow^{\mathrlap{(f, Id)}} && \downarrow^\mathrlap{f} \\ Path(Y) &\overset{(p_1,p_0) \in Fib }{\longrightarrow}& Y \times Y &\stackrel{pr_1}{\longrightarrow}& Y \\ {}^{\mathllap{p_0}}\downarrow & \swarrow_{\mathrlap{pr_2 \atop {\in Fib}}} \\ Y } \,.

Both squares are pullback squares. Since pullbacks of fibrations are fibrations by prop. , the morphism Path(f)X×YPath(f) \to X \times Y is a fibration. Similarly, since XX is fibrant, also the projection map X×YYX \times Y \to Y is a fibration (being the pullback of X*X \to \ast along Y*Y \to \ast).

Since the vertical composite is thereby exhibited as the composite of two fibrations

Path(f)(f,id) *(p 1,p 0)X×Ypr 2(f,Id)=pr 2Y, Path(f) \overset{(f,id)^\ast(p_1,p_0)}{\longrightarrow} X \times Y \stackrel{pr_2 \circ (f ,Id) = pr_2}{\longrightarrow} Y \,,

it is itself a fibration.

Then to see that there is a weak equivalence as claimed:

The universal property of the pullback Path(f)Path(f) induces a right inverse of Path(f)XPath(f) \to X fitting into this diagram

id X: X W Path(f) WFib X f f id Y: Y Wi Path(Y) p 1 Y Id p 0 Y, \array{ id_X \colon & X &\underoverset{\in W}{\exists}{\longrightarrow} & Path(f) & \overset{\in W \cap Fib}{\longrightarrow}& X \\ & {}^{\mathrlap{f}}\downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ id_Y\colon& Y &\underoverset{\in W}{i}{\longrightarrow}& Path(Y) &\stackrel{p_1}{\to}& Y \\ & & {}_{\mathllap{Id}}\searrow& \downarrow^{\mathrlap{p_0}} \\ & && Y } \,,

which is a weak equivalence, as indicated, by two-out-of-three (def. ).

This establishes the claim.

Categories of fibrant objects

Below we discuss the homotopy-theoretic properties of the mapping cone- and mapping cocone-constructions from above. Before we do so, we here establish a collection of general facts that hold in categories of fibrant objects and dually in categories of cofibrant objects, def. .

Literature (Brown 73, section 4).

Lemma

Let f:XYf\colon X \longrightarrow Y be a morphism in a category of fibrant objects, def. . Then given any choice of path space objects Path(X)Path(X) and Path(Y)Path(Y), def. , there is a replacement of Path(X)Path(X) by a path space object Path(X)˜\widetilde{Path(X)} along an acylic fibration, such that Path(X)˜\widetilde{Path(X)} has a morphism ϕ\phi to Path(Y)Path(Y) which is compatible with the structure maps, in that the following diagram commutes

X f Y Path(X) WFib Path(X)˜ ϕ Path(Y) (p 0 X,p 1 X) (p 0 Y,p 1 Y) (p˜ 0 X,p˜ 1 X) X×X (f,f) Y×Y. \array{ && X &\overset{f}{\longrightarrow}& Y \\ &\swarrow& \downarrow && \downarrow \\ Path(X) &\underset{\in W \cap Fib}{\longleftarrow}& \widetilde{Path(X)} &\overset{\phi}{\longrightarrow}& Path(Y) \\ &{}_{\mathllap{(p^X_0,p^X_1)}}\searrow& \downarrow^{\mathrlap{(p^Y_0,p^Y_1)}} && \downarrow^{\mathrlap{(\tilde p^X_0,\tilde p^X_1)}} \\ && X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,.

(Brown 73, section 2, lemma 2)

Proof

Consider the commuting square

X f Y Path(Y) (p 0 Y,p 1 Y) Path(X) (p 0 X,p 1 X) X×X (f,f) Y×Y. \array{ X &\overset{f}{\longrightarrow}& Y &\longrightarrow& Path(Y) \\ \downarrow && && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ Path(X) &\overset{(p^X_0,p^X_1)}{\longrightarrow}& X \times X &\overset{(f,f)}{\longrightarrow}& Y \times Y } \,.

Then consider its factorization through the pullback of the right morphism along the bottom morphism,

X (fp 0 X,fp 1 X) *Path(Y) Path(Y) W WFib Fib (p 0 Y,p 1 Y) Path(X) (fp 0 X,fp 1 X) Y×Y. \array{ X &\longrightarrow& (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y) &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}}_{\mathrlap{\in Fib}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,.

Finally use the factorization lemma to factor the morphism X(fp 0 X,fp 1 X) *Path(Y)X \to (f \circ p_0^X, f\circ p_1^X)^\ast Path(Y) through a weak equivalence followed by a fibration, the object this factors through serves as the desired path space resolution

X W Path(X)˜ Path(Y) W WFib (p 0 Y,p 1 Y) Path(X) (fp 0 X,fp 1 X) Y×Y. \array{ X &\overset{\in W}{\longrightarrow}& \widetilde{Path(X)} &\longrightarrow& Path(Y) \\ &{}_{\mathllap{\in W}}\searrow& \downarrow^{\mathrlap{\in W \cap Fib}} && \downarrow^{\mathrlap{(p_0^Y, p_1^Y)}} \\ && Path(X) &\overset{(f \circ p_0^X, f\circ p_1^X)}{\longrightarrow}& Y \times Y } \,.
Lemma

In a category of fibrant objects 𝒞 f\mathcal{C}_f, def. , let

A 1 f A 2 Fib Fib B \array{ A_1 &&\stackrel{f}{\longrightarrow}&& A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B }

be a morphism over some object BB in 𝒞 f\mathcal{C}_f and let u:BBu \colon B' \to B be any morphism in 𝒞 f\mathcal{C}_f. Let

u *A 1 u *f u *A 2 Fib Fib B \array{ u^*A_1 &&\stackrel{u^* f}{\longrightarrow}&& u^* A_2 \\ & {}_{\in Fib}\searrow && \swarrow_{\in Fib} \\ && B' }

be the corresponding morphism pulled back along uu.

Then

  • if ff is a fibration then also u *fu^* f is a fibration;

  • if ff is a weak equivalence then also u *fu^* f is a weak equivalence.

(Brown 73, section 4, lemma 1)

Proof

For fFibf \in Fib the statement follows from the pasting law which says that if in

B× BA 1 A 1 u *fFib fFib B× BA 2 A 2 Fib Fib B u B \array{ B' \times_B A_1 &\longrightarrow& A_1 \\ \;\;\downarrow^{\mathrlap{u^* f \in Fib}} && \;\;\downarrow^{\mathrlap{f \in Fib}} \\ B' \times_B A_2 &\longrightarrow& A_2 \\ \;\downarrow^{\mathrlap{\in Fib}} && \;\downarrow^{\mathrlap{\in Fib}} \\ B' &\stackrel{u}{\longrightarrow}& B }

the bottom and the total square are pullback squares, then so is the top square. The same reasoning applies for fWFibf \in W \cap Fib.

Now to see the case that fWf\in W:

Consider the full subcategory (𝒞 /B) f(\mathcal{C}_{/B})_f of the slice category 𝒞 /B\mathcal{C}_{/B} (def. ) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations

X Fib p B \array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B }

into BB. By factorizing for every such fibration the diagonal morphisms into the fiber product X×BXX \underset{B}{\times} X through a weak equivalence followed by a fibration, we obtain path space objects Path B(X)Path_B(X) relative to BB:

(Δ X)/B: X W Path B(X) Fib X×BX Fib Fib B. \array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.

With these, the factorization lemma (lemma ) applies in (𝒞 /B) f(\mathcal{C}_{/B})_f.

(Notice that for this we do need the restriction of 𝒞 /B\mathcal{C}_{/B} to the fibrations, because this ensures that the projections p i:X 1× BX 2X ip_i \colon X_1 \times_B X_2 \to X_i are still fibrations, which is used in the proof of the factorization lemma (here).)

So now given any

X Wf Y Fib Fib B \array{ X && \underoverset{\in W}{f}{\longrightarrow} && Y \\ & {}_{\mathllap{\in Fib}}\searrow && \swarrow_{\mathrlap{\in Fib}} \\ && B }

apply the factorization lemma in (𝒞 /B) f(\mathcal{C}_{/B})_f to factor it as

X iW Path B(f) WFib Y Fib Fib B. \array{ X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& Y \\ & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ && B } \,.

By the previous discussion it is sufficient now to show that the base change of ii to BB' is still a weak equivalence. But by the factorization lemma in (𝒞 /B) f(\mathcal{C}_{/B})_f, the morphism ii is right inverse to another acyclic fibration over BB:

id X: X iW Path B(f) WFib X Fib Fib B. \array{ id_X \;\colon & X &\overset{i \in W}{\longrightarrow}& Path_B(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ & & {}_{\mathllap{\in Fib}}\searrow &\downarrow& \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.

(Notice that if we had applied the factorization lemma of Δ X\Delta_X in 𝒞 f\mathcal{C}_f instead of (Δ X)/B(\Delta_X)/B in (𝒞 /B)(\mathcal{C}_{/B}) then the corresponding triangle on the right here would not commute.)

Now we may reason as before: the base change of the top morphism here is exhibited by the following pasting composite of pullbacks:

B×BX X (pb) B×BPath B(f) Path B(f) (pb) WFib B×BX X (pb) B B. \array{ B' \underset{B}{\times} X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' \underset{B}{\times} Path_B(f) &\longrightarrow& Path_B(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in W \cap Fib}} \\ B' \underset{B}{\times}X &\longrightarrow& X \\ \downarrow &(pb)& \downarrow \\ B' &\longrightarrow& B } \,.

The acyclic fibration Path B(f)Path_B(f) is preserved by this pullback, as is the identity id X:XPath B(X)Xid_X \colon X \to Path_B(X)\to X. Hence the weak equivalence XPath B(X)X \to Path_B(X) is preserved by two-out-of-three (def. ).

Lemma

In a category of fibrant objects, def. , the pullback of a weak equivalence along a fibration is again a weak equivalence.

(Brown 73, section 4, lemma 2)

Proof

Let u:BBu \colon B' \to B be a weak equivalence and p:EB p \colon E \to B be a fibration. We want to show that the left vertical morphism in the pullback

E× BB B W W E Fib B \array{ E \times_B B' &\longrightarrow& B' \\ \downarrow^{\mathrlap{\Rightarrow \in W} } && \;\downarrow^{\mathrlap{\in W}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B }

is a weak equivalence.

First of all, using the factorization lemma we may factor BBB' \to B as

BWPath(u)WFB B' \stackrel{\in W}{\longrightarrow} Path(u) \stackrel{\in W \cap F}{\longrightarrow} B

with the first morphism a weak equivalence that is a right inverse to an acyclic fibration and the right one an acyclic fibration.

Then the pullback diagram in question may be decomposed into two consecutive pullback diagrams

E× BB B Q Fib Path(u) WFib WFib E Fib B, \array{ E \times_B B' &\to& B' \\ \downarrow && \downarrow \\ Q &\stackrel{\in Fib}{\to}& Path(u) \\ \;\;\downarrow^{\mathrlap{\in W \cap Fib}} && \;\;\downarrow^{\mathrlap{\in W \cap Fib}} \\ E &\stackrel{\in Fib}{\longrightarrow}& B } \,,

where the morphisms are indicated as fibrations and acyclic fibrations using the stability of these under arbitrary pullback.

This means that the proof reduces to proving that weak equivalences u:BWBu : B' \stackrel{\in W}{\to} B that are right inverse to some acyclic fibration v:BWFBv : B \stackrel{\in W \cap F}{\to} B' map to a weak equivalence under pullback along a fibration.

Given such uu with right inverse vv, consider the pullback diagram

E (p,id)W id E 1 B× BE WFib E Fib pFib (pb) B vWFib B vFibW B. \array{ & E \\ & {}^{\mathllap{{(p,id)}\atop \in W}}\downarrow & \searrow^{\mathrlap{id}} \\ E_1 \coloneqq & B \times_{B'} E & \stackrel{\in W \cap Fib }{\longrightarrow} & E \\ & \downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib }} \\ & &(pb)& B \\ & \downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ & B &\overset{v \in Fib \cap W}{\longrightarrow}& B' } \,.

Notice that the indicated universal morphism p×Id:EWE 1p \times Id \colon E \stackrel{\in W}{\to} E_1 into the pullback is a weak equivalence by two-out-of-three (def. ).

The previous lemma says that weak equivalences between fibrations over BB are themselves preserved by base extension along u:BBu \colon B' \to B. In total this yields the following diagram

u *E=B× BE E u *(p×Id)W p×IdW id u *E 1 E 1 WFib E Fib Fib pFib B vWFib B u B vWFib B \array{ && u^* E = B' \times_B E &\longrightarrow & E \\ && {}^{\mathllap{ {u^*(p \times Id)} \atop {\in W} }}\downarrow && {}^{\mathllap{ {p \times Id} \atop {\in W} }}\downarrow & \searrow^{\mathrlap{id}} \\ && u^* E_1 &\longrightarrow& E_1 &\stackrel{\in W \cap Fib}{\longrightarrow}& E \\ &&\downarrow^{\mathrlap{\in Fib}}&&\downarrow^{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p \in Fib}} \\ &&&&&& B \\ &&\downarrow&&\downarrow && \downarrow^{\mathrlap{v \in W \cap Fib}} \\ && B' &\stackrel{u}{\longrightarrow}& B &\stackrel{v \in W \cap Fib}{\longrightarrow}& B' }

so that with p×Id:EE 1p \times Id : E \to E_1 a weak equivalence also u *(p×Id)u^* (p \times Id) is a weak equivalence, as indicated.

Notice that u *E=B× BEEu^* E = B' \times_B E \to E is the morphism that we want to show is a weak equivalence. By two-out-of-three (def. ) for that it is now sufficient to show that u *E 1E 1u^* E_1 \to E_1 is a weak equivalence.

That finally follows now since, by assumption, the total bottom horizontal morphism is the identity. Hence so is the top horizontal morphism. Therefore u *E 1E 1u^\ast E_1 \to E_1 is right inverse to a weak equivalence, hence is a weak equivalence.

Lemma

Let (𝒞 */) f(\mathcal{C}^{\ast/})_f be a category of fibrant objects, def. in a model structure on pointed objects (prop. ). Given any commuting diagram in 𝒞 \mathcal{C}^{} of the form

X 1 tW X 1 gf X 2 Fib p 1 Fib p 2 B u C \array{ X'_1 &\underoverset{t}{\in W}{\longrightarrow}& X_1 &\stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}}& X_2 \\ && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in Fib}} && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in Fib}} \\ && B &\overset{u}{\longrightarrow}& C }

(meaning: both squares commute and tt equalizes ff with gg) then the localization functor γ:(𝒞 */) fHo(𝒞 */)\gamma \colon (\mathcal{C}^{\ast/})_f \to Ho(\mathcal{C}^{\ast/}) (def. , cor ) takes the morphisms fib(p 1)fib(p 2)fib(p_1) \stackrel{\longrightarrow}{\longrightarrow} fib(p_2) induced by ff and gg on fibers (example ) to the same morphism, in the homotopy category.

(Brown 73, section 4, lemma 4)

Proof

First consider the pullback of p 2p_2 along uu: this forms the same kind of diagram but with the bottom morphism an identity. Hence it is sufficient to consider this special case.

Consider the full subcategory (𝒞 /B */) f(\mathcal{C}^{\ast/}_{/B})_f of the slice category 𝒞 /B */\mathcal{C}^{\ast/}_{/B} (def. ) on its fibrant objects, i.e. the full subcategory of the slice category on the fibrations

X Fib p B \array{ X \\ \downarrow^{\mathrlap{p}}_{\mathrlap{\in Fib}} \\ B }

into BB. By factorizing for every such fibration the diagonal morphisms into the fiber product X×BXX \underset{B}{\times} X through a weak equivalence followed by a fibration, we obtain path space objects Path B(X)Path_B(X) relative to BB:

(Δ X)/B: X W Path B(X) Fib X×BX Fib Fib B. \array{ (\Delta_X)/B \;\colon & X &\overset{\in W}{\longrightarrow}& Path_B(X) &\overset{\in Fib}{\longrightarrow}& X \underset{B}{\times} X \\ & & {}_{\mathllap{\in Fib}}\searrow & \downarrow & \swarrow_{\mathrlap{\in Fib}} \\ & && B } \,.

With these, the factorization lemma (lemma ) applies in (𝒞 /B */) f(\mathcal{C}^{\ast/}_{/B})_f.

Let then XsPath B(X 2)(p 0,p 1)X 2× BX 2X\overset{s}{\to}Path_B(X_2)\overset{(p_0,p_1)}{\to} X_2 \times_B X_2 be a path space object for X 2X_2 in the slice over BB and consider the following commuting square

X 1 sft Path B(X 2) W t Fib (p 0,p 1) X 1 (f,g) X 2×BX 2. \array{ X'_1 &\overset{s f t}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,.

By factoring this through the pullback (f,g) *(p 0,p 1)(f,g)^\ast(p_0,p_1) and then applying the factorization lemma and then two-out-of-three (def. ) to the factoring morphisms, this may be replaced by a commuting square of the same form, where however the left morphism is an acyclic fibration

X 1 Path B(X 2) WFib t Fib (p 0,p 1) X 1 (f,g) X 2×BX 2. \array{ X''_1 &\overset{}{\longrightarrow}& Path_B(X_2) \\ {}^{\mathllap{t}}_{\mathllap{\in W\cap Fib}} \downarrow && \downarrow^{\mathrlap{(p_0,p_1)}}_{\mathrlap{\in Fib}} \\ X_1 &\overset{(f,g)}{\longrightarrow}& X_2\underset{B}{\times} X_2 } \,.

This makes also the morphism X 1BX''_1 \to B be a fibration, so that the whole diagram may now be regarded as a diagram in the category of fibrant objects (𝒞 /B) f(\mathcal{C}_{/B})_f of the slice category over BB.

As such, the top horizontal morphism now exhibits a right homotopy which under localization γ B:(𝒞 /B) fHo(𝒞 /B)\gamma_B \;\colon\; (\mathcal{C}_{/B})_f \longrightarrow Ho(\mathcal{C}_{/B}) (def. ) of the slice model structure (prop. ) we have

γ B(f)=γ B(g). \gamma_B(f) = \gamma_B(g) \,.

The result then follows by observing that we have a commuting square of functors

(𝒞 /B */) f fib 𝒞 */ γ B γ Ho(𝒞 /B */) Ho(𝒞 */), \array{ (\mathcal{C}^{\ast/}_{/B})_f &\overset{fib}{\longrightarrow}& \mathcal{C}^{\ast/} \\ \downarrow^{\mathrlap{\gamma_B}} &\swArrow& \downarrow^{\mathrlap{\gamma}} \\ Ho(\mathcal{C}^{\ast/}_{/B}) &\longrightarrow& Ho(\mathcal{C}^{\ast/}) } \,,

because, by lemma , the top and right composite sends weak equivalences to isomorphisms, and hence the bottom filler exists by theorem . This implies the claim.

Homotopy fibers

We now discuss the homotopy-theoretic properties of the mapping cone- and mapping cocone-constructions from above.

Literature (Brown 73, section 4).

Remark

The factorization lemma with prop. says that the mapping cocone of a morphism ff, def. , is equivalently the plain fiber, example , of a fibrant resolution f˜\tilde f of ff:

Path *(f) Path(f) (pb) f˜ * Y. \array{ Path_\ast(f) &\longrightarrow& Path(f) \\ \downarrow &(pb)& \downarrow^{\mathrlap{\tilde f}} \\ \ast &\longrightarrow& Y } \,.

The following prop. says that, up to equivalence, this situation is independent of the specific fibration resolution f˜\tilde f provided by the factorization lemma (hence by the prescription for the mapping cocone), but only depends on it being some fibration resolution.

Proposition

In the category of fibrant objects (𝒞 */) f(\mathcal{C}^{\ast/})_f, def. , of a model structure on pointed objects (prop. ) consider a morphism of fiber-diagrams, hence a commuting diagram of the form

fib(p 1) X 1 Fibp 1 Y 1 h g f fib(p 2) X 2 Fibp 2 Y 2. \array{ fib(p_1) &\longrightarrow& X_1 &\underoverset{\in Fib}{p_1}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{h}} && \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 } \,.

If ff and gg weak equivalences, then so is hh.

Proof

Factor the diagram in question through the pullback of p 2p_2 along ff

fib(p 1) X 1 h W p 1 fib(f *p 2) f *X 2 Fibf *p 2 Y 1 W W f fib(p 2) X 2 Fibp 2 Y 2 \array{ fib(p_1) &\longrightarrow& X_1 \\ \downarrow^{\mathrlap{h}} && {}^{\mathllap{\in W}}\downarrow & \searrow^{\mathrlap{p_1}} & \\ fib(f^\ast p_2) &\longrightarrow& f^\ast X_2 &\underoverset{\in Fib}{f^\ast p_2}{\longrightarrow}& Y_1 \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in W}} \\ fib(p_2) &\longrightarrow& X_2 &\underoverset{\in Fib}{p_2}{\longrightarrow}& Y_2 }

and observe that

  1. fib(f *p 2)=pt *f *p 2=pt *p 2=fib(p 2)fib(f^\ast p_2) = pt^\ast f^\ast p_2 = pt^\ast p_2 = fib(p_2);

  2. f *X 2X 2f^\ast X_2 \to X_2 is a weak equivalence by lemma ;

  3. X 1f *X 2X_1 \to f^\ast X_2 is a weak equivalence by assumption and by two-out-of-three (def. );

Moreover, this diagram exhibits h:fib(p 1)fib(f *p 2)=fib(p 2)h \colon fib(p_1)\to fib(f^\ast p_2) = fib(p_2) as the base change, along *Y 1\ast \to Y_1, of X 1f *X 2X_1 \to f^\ast X_2. Therefore the claim now follows with lemma .

Hence we say:

Definition

Let 𝒞\mathcal{C} be a model category and 𝒞 */\mathcal{C}^{\ast/} its model category of pointed objects, prop. . For f:XYf \colon X \longrightarrow Y any morphism in its category of fibrant objects (𝒞 */) f(\mathcal{C}^{\ast/})_f, def. , then its homotopy fiber

hofib(f)X hofib(f)\longrightarrow X

is the morphism in the homotopy category Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}), def. , which is represented by the fiber, example , of any fibration resolution f˜\tilde f of ff (hence any fibration f˜\tilde f such that ff factors through a weak equivalence followed by f˜\tilde f).

Dually:

For f:XYf \colon X \longrightarrow Y any morphism in its category of cofibrant objects (𝒞 */) c(\mathcal{C}^{\ast/})_c, def. , then its homotopy cofiber

Yhocofib(f) Y \longrightarrow hocofib(f)

is the morphism in the homotopy category Ho(𝒞)Ho(\mathcal{C}), def. , which is represented by the cofiber, example , of any cofibration resolution of ff (hence any cofibration f˜\tilde f such that ff factors as f˜\tilde f followed by a weak equivalence).

Proposition

The homotopy fiber in def. is indeed well defined, in that for f 1f_1 and f 2f_2 two fibration replacements of any morphisms ff in 𝒞 f\mathcal{C}_f, then their fibers are isomorphic in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}).

Proof

It is sufficient to exhibit an isomorphism in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}) from the fiber of the fibration replacement given by the factorization lemma (for any choice of path space object) to the fiber of any other fibration resolution.

Hence given a morphism f:YXf \colon Y \longrightarrow X and a factorization

f:XWX^f 1FibY f \;\colon\; X \underset{\in W}{\longrightarrow} \hat X \underoverset{f_1}{\in Fib}{\longrightarrow} Y

consider, for any choice Path(Y)Path(Y) of path space object (def. ), the diagram

Path(f) WFib X W (pb) W Path(f 1) WFib X^ Fib (pb) f 1Fib Path(Y) WFibp 1 Y p 0WFib Y \array{ Path(f) &\overset{\in W \cap Fib}{\longrightarrow}& X \\ {}^{\mathllap{\in W}}\downarrow &(pb)& \downarrow^{\mathrlap{\in W}} \\ Path(f_1) &\overset{\in W \cap Fib}{\longrightarrow}& \hat X \\ {}^{\mathllap{\in Fib}}\downarrow &(pb)& \downarrow^{\mathrlap{ {f_1} \atop {\in Fib}}} \\ Path(Y) &\underoverset{\in W \cap Fib}{p_1}{\longrightarrow}& Y \\ {}^{\mathllap{ {p_0} \atop \in W \cap Fib}}\downarrow \\ Y }

as in the proof of lemma . Now by repeatedly using prop. :

  1. the bottom square gives a weak equivalence from the fiber of Path(f 1)Path(Y)Path(f_1) \to Path(Y) to the fiber of f 1f_1;

  2. The square

    Path(f 1) id Path(f 1) Path(Y) p 0 Y \array{ Path(f_1) &\overset{id}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Path(Y) &\underset{p_0}{\longrightarrow}& Y }

    gives a weak equivalence from the fiber of Path(f 1)Path(Y)Path(f_1) \to Path(Y) to the fiber of Path(f 1)YPath(f_1)\to Y.

  3. Similarly the total vertical composite gives a weak equivalence via

    Path(f) W Path(f 1) Y id Y \array{ Path(f) &\overset{\in W}{\longrightarrow}& Path(f_1) \\ \downarrow && \downarrow \\ Y &\underset{id}{\longrightarrow}& Y }

from the fiber of Path(f)YPath(f) \to Y to the fiber of Path(f 1)YPath(f_1)\to Y.

Together this is a zig-zag of weak equivalences of the form

fib(f 1)Wfib(Path(f 1)Path(Y))Wfib(Path(f 1)Y)Wfib(Path(f)Y) fib(f_1) \;\overset{\in W}{\longleftarrow}\; fib(Path(f_1)\to Path(Y)) \;\overset{\in W}{\longrightarrow}\; fib(Path(f_1)\to Y) \;\overset{\in W}{\longleftarrow}\; fib(Path(f) \to Y)

between the fiber of Path(f)YPath(f) \to Y and the fiber of f 1f_1. This gives an isomorphism in the homotopy category.

Example

(fibers of Serre fibrations)

In showing that Serre fibrations are abstract fibrations in the sense of model category theory, theorem implies that the fiber FF (example ) of a Serre fibration, def.

F X p B \array{ F &\longrightarrow& X \\ && \downarrow^{\mathrlap{p}} \\ && B }

over any point is actually a homotopy fiber in the sense of def. . With prop. this implies that the weak homotopy type of the fiber only depends on the Serre fibration up to weak homotopy equivalence in that if p:XBp' \colon X' \to B' is another Serre fibration fitting into a commuting diagram of the form

X W cl X p p B W cl B \array{ X &\overset{\in W_{cl}}{\longrightarrow}& X' \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{p'}} \\ B &\overset{\in W_{cl}}{\longrightarrow}& B' }

then FW clFF \overset{\in W_{cl}}{\longrightarrow} F'.

In particular this gives that the weak homotopy type of the fiber of a Serre fibration p:XBp \colon X \to B does not change as the basepoint is moved in the same connected component. For let γ:IB\gamma \colon I \longrightarrow B be a path between two points

b 0,1:*W cli 0,1IγB. b_{0,1} \;\colon\; \ast \underoverset{\in W_{cl}}{i_{0,1}}{\longrightarrow} I \overset{\gamma}{\longrightarrow} B \,.

Then since all objects in (Top cg) Quillen(Top_{cg})_{Quillen} are fibrant, and since the endpoint inclusions i 0,1i_{0,1} are weak equivalences, lemma gives the zig-zag of top horizontal weak equivalences in the following diagram:

F b 0= b 0 *p W cl γ *p W cl b 1 *p =F b 1 (pb) γ *fFib (pb) * i 0W cl I i 1W cl * \array{ F_{b_0} = & b_0^\ast p &\overset{\in W_{cl}}{\longrightarrow}& \gamma^{\ast}p &\overset{\in W_{cl}}{\longleftarrow}& b_1^\ast p & = F_{b_1} \\ & \downarrow &(pb)& \downarrow{\mathrlap{{\gamma^\ast f} \atop {\in \atop {Fib}}}} &\;\;(pb)& \downarrow \\ & \ast &\underoverset{i_0}{\in W_{cl}}{\longrightarrow}& I &\underoverset{i_1}{\in W_{cl}}{\longleftarrow}& \ast }

and hence an isomorphism F b 0F b 1F_{b_0} \simeq F_{b_1} in the classical homotopy category (def. ).

The same kind of argument applied to maps from the square I 2I^2 gives that if γ 1,γ 2:IB\gamma_1, \gamma_2\colon I \to B are two homotopic paths with coinciding endpoints, then the isomorphisms between fibers over endpoints which they induce are equal. (But in general the isomorphism between the fibers does depend on the choice of homotopy class of paths connecting the basepoints!)

The same kind of argument also shows that if BB has the structure of a cell complex (def. ) then the restriction of the Serre fibration to one cell D nD^n may be identified in the homotopy category with D n×FD^n \times F, and may be canonically identified so if the fundamental group of XX is trivial. This is used when deriving the Serre-Atiyah-Hirzebruch spectral sequence for pp (prop.).

Example

For every continuous function f:XYf \colon X \longrightarrow Y between CW-complexes, def. , then the standard topological mapping cone is the attaching space (example )

Y fCone(X)Top Y \cup_f Cone(X) \;\; \in Top

of YY with the standard cone Cone(X)Cone(X) given by collapsing one end of the standard topological cyclinder X×IX \times I (def. ) as shown in example .

Equipped with the canonical continuous function

YY fCone(X) Y \longrightarrow Y \cup_f Cone(X)

this represents the homotopy cofiber, def. , of ff with respect to the classical model structure on topological spaces 𝒞=Top Quillen\mathcal{C}= Top_{Quillen} from theorem .

Proof

By prop. , for XX a CW-complex then the standard topological cylinder object X×IX\times I is indeed a cyclinder object in Top QuillenTop_{Quillen}. Therefore by prop. and the factorization lemma , the mapping cone construction indeed produces first a cofibrant replacement of ff and then the ordinary cofiber of that, hence a model for the homotopy cofiber.

Example

The homotopy fiber of the inclusion of classifying spaces BO(n)BO(n+1)B O(n) \hookrightarrow B O(n+1) is the n-sphere S nS^n. See this prop. at Classifying spaces and G-structure.

Example

Suppose a morphism f:XYf \colon X \longrightarrow Y already happens to be a fibration between fibrant objects. The factorization lemma replaces it by a fibration out of the mapping cocylinder Path(f)Path(f), but such that the comparison morphism is a weak equivalence:

fib(f) X Fibf Y W W id fib(f˜) Path(f) Fibf˜ Y. \array{ fib(f) &\longrightarrow& X &\underoverset{\in Fib}{f}{\longrightarrow}& Y \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{id}} \\ fib(\tilde f) &\longrightarrow& Path(f) &\underoverset{\in Fib}{\tilde f}{\longrightarrow}& Y } \,.

Hence by prop. in this case the ordinary fiber of ff is weakly equivalent to the mapping cocone, def. .

We may now state the abstract version of the statement of prop. :

Proposition

Let 𝒞\mathcal{C} be a model category. For f:XYf \colon X \to Y any morphism of pointed objects, and for AA a pointed object, def. , then the sequence

[A,hofib(f)] *i *[A,X] *f *[A,Y] * [A,hofib(f)]_\ast \overset{i_\ast}{\longrightarrow} [A,X]_\ast \overset{f_\ast}{\longrightarrow} [A,Y]_{\ast}

is exact as a sequence of pointed sets.

(Where the sequence here is the image of the homotopy fiber sequence of def. under the hom-functor [A,] *:Ho(𝒞 */)Set */[A,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/} from example .)

Proof

Let AA, XX and YY denote fibrant-cofibrant objects in 𝒞 */\mathcal{C}^{\ast/} representing the given objects of the same name in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}). Moreover, let ff be a fibration in 𝒞 */\mathcal{C}^{\ast/} representing the given morphism of the same name in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}).

Then by def. and prop. there is a representative hofib(f)𝒞hofib(f) \in \mathcal{C} of the homotopy fiber which fits into a pullback diagram of the form

hofib(f) i X f * Y \array{ hofib(f) &\overset{i}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }

With this the hom-sets in question are represented by genuine morphisms in 𝒞 */\mathcal{C}^{\ast/}, modulo homotopy. From this it follows immediately that im(i *)im(i_\ast) includes into ker(f *)ker(f_\ast). Hence it remains to show the converse: that every element in ker(f *)ker(f_\ast) indeed comes from im(i *)im(i_\ast).

But an element in ker(f *)ker(f_\ast) is represented by a morphism α:AX\alpha \colon A \to X such that there is a left homotopy as in the following diagram

A α X i 0 η˜ f A i 1 Cyl(A) η Y = * Y. \array{ && A &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{i_0}}\downarrow &{}^{\tilde \eta}\nearrow& \downarrow^{\mathrlap{f}} \\ A &\overset{i_1}{\longrightarrow} & Cyl(A) &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow^{\mathrlap{=}} \\ \ast && \longrightarrow && Y } \,.

Now by lemma the square here has a lift η˜\tilde \eta, as shown. This means that i 1η˜i_1 \circ\tilde \eta is left homotopic to α\alpha. But by the universal property of the fiber, i 1η˜i_1 \circ \tilde \eta factors through i:hofib(f)Xi \colon hofib(f) \to X.

With prop. it also follows notably that the loop space construction becomes well-defined on the homotopy category:

Remark

Given an object X𝒞 f */X \in \mathcal{C}^{\ast/}_f, and picking any path space object Path(X)Path(X), def. with induced loop space object ΩX\Omega X, def. , write Path 2(X)=Path(X)×XPath(X)Path_2(X) = Path(X) \underset{X}{\times} Path(X) for the path space object given by the fiber product of Path(X)Path(X) with itself, via example . From the pullback diagram there, the fiber inclusion ΩXPath(X)\Omega X \to Path(X) induces a morphism

ΩX×ΩX(ΩX) 2. \Omega X \times \Omega X \longrightarrow (\Omega X)_2 \,.

In the case where 𝒞 */=Top */\mathcal{C}^{\ast/} = Top^{\ast/} and Ω\Omega is induced, via def. , from the standard path space object (def. ), i.e. in the case that

ΩX=fib(Maps(I +,X) *X×X), \Omega X = fib(Maps(I_+,X)_\ast \longrightarrow X \times X) \,,

then this is the operation of concatenating two loops parameterized by I=[0,1]I = [0,1] to a single loop parameterized by [0,2][0,2].

Proposition

Let 𝒞\mathcal{C} be a model category, def. . Then the construction of forming loop space objects XΩXX\mapsto \Omega X, def. (which on 𝒞 f */\mathcal{C}^{\ast/}_f depends on a choice of path space objects, def. ) becomes unique up to isomorphism in the homotopy category (def. ) of the model structure on pointed objects (prop. ) and extends to a functor:

Ω:Ho(𝒞 */)Ho(𝒞 */). \Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.

Dually, the reduced suspension operation, def. , which on 𝒞 */\mathcal{C}^{\ast/} depends on a choice of cylinder object, becomes a functor on the homotopy category

Σ:Ho(𝒞 */)Ho(𝒞 */). \Sigma \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.

Moreover, the pairing operation induced on the objects in the image of this functor via remark (concatenation of loops) gives the objects in the image of Ω\Omega group object structure, and makes this functor lift as

Ω:Ho(𝒞 */)Grp(Ho(𝒞 */)). \Omega \;\colon\; Ho(\mathcal{C}^{\ast/}) \longrightarrow Grp(Ho(\mathcal{C}^{\ast/})) \,.

(Brown 73, section 4, theorem 3)

Proof

Given an object X𝒞 */X \in \mathcal{C}^{\ast/} and given two choices of path space objects Path(X)Path(X) and Path(X)˜\widetilde{Path(X)}, we need to produce an isomorphism in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}) between ΩX\Omega X and Ω˜X\tilde \Omega X.

To that end, first lemma implies that any two choices of path space objects are connected via a third path space by a span of morphisms compatible with the structure maps. By two-out-of-three (def. ) every morphism of path space objects compatible with the inclusion of the base object is a weak equivalence. With this, lemma implies that these morphisms induce weak equivalences on the corresponding loop space objects. This shows that all choices of loop space objects become isomorphic in the homotopy category.

Moreover, all the isomorphisms produced this way are actually equal: this follows from lemma applied to

X s Path(X) Path(X)˜ X×X id X×X. \array{ X &\overset{s}{\longrightarrow}& Path(X) &\stackrel{\longrightarrow}{\longrightarrow}& \widetilde{Path(X)} \\ && \downarrow && \downarrow \\ && X\times X &\overset{id}{\longrightarrow}& X \times X } \,.

This way we obtain a functor

Ω:𝒞 f */Ho(𝒞 */). \Omega \;\colon\; \mathcal{C}^{\ast/}_f \longrightarrow Ho(\mathcal{C}^{\ast/}) \,.

By prop. (and using that Cartesian product preserves weak equivalences) this functor sends weak equivalences to isomorphisms. Therefore the functor on homotopy categories now follows with theorem .

It is immediate to see that the operation of loop concatenation from remark gives the objects ΩXHo(𝒞 */)\Omega X \in Ho(\mathcal{C}^{\ast/}) the structure of monoids. It is now sufficient to see that these are in fact groups:

We claim that the inverse-assigning operation is given by the left map in the following pasting composite

ΩX Path(X) X×X (pb) swap ΩX Path(X) (p 0,p 1) X×X, \array{ \Omega' X &\longrightarrow& Path'(X) &\overset{}{\longrightarrow}& X \times X \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} &(pb)& \downarrow^{\mathrlap{swap}} \\ \Omega X &\longrightarrow& Path(X) &\underset{(p_0,p_1)}{\longrightarrow}& X \times X } \,,

(where Path(X)Path'(X), thus defined, is the path space object obtained from Path(X)Path(X) by “reversing the notion of source and target of a path”).

To see that this is indeed an inverse, it is sufficient to see that the two morphisms

ΩX(ΩX) 2 \Omega X \stackrel{\longrightarrow}{\longrightarrow} (\Omega X)_2

induced from

Path(X)(sp 0,sp 0)ΔPath(X)× XPath(X) \array{ Path(X) \stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}} Path(X) \times_X Path'(X) }

coincide in the homotopy category. This follows with lemma applied to the following commuting diagram:

X i Path(X) (sp 0,sp 0)Δ Path(X)× XPath(X) (p 0,p 1) X×X Δpr 1 X×X. \array{ X &\overset{i}{\longrightarrow}& Path(X) &\stackrel{\overset{\Delta}{\longrightarrow}}{\underset{(s\circ p_0,s \circ p_0)}{\longrightarrow}}& Path(X)\times_X Path'(X) \\ && {}^{\mathllap{(p_0,p_1)}}\downarrow && \downarrow^{\mathrlap{}} \\ && X\times X &\overset{\Delta \circ pr_1}{\longrightarrow}& X \times X } \,.

Homotopy pullbacks

The concept of homotopy fibers of def. is a special case of the more general concept of homotopy pullbacks.

Definition

(proper model category)

A model category 𝒞\mathcal{C} (def. ) is called

Example

By lemma , a model category 𝒞\mathcal{C} (def. ) in which all objects are fibrant is a right proper model category (def. ).

Definition

Let 𝒞\mathcal{C} be a right proper model category (def. ). Then a commuting square

A B g C f D \array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{g}} \\ C &\underset{f}{\longrightarrow}& D }

in 𝒞 f\mathcal{C}_f is called a homotopy pullback (of ff along gg and equivalently of gg along ff) if the following equivalent conditions hold:

  1. for some factorization of the form

    g:BWB^FibD g \colon B \overset{\in W }{\longrightarrow} \hat B \overset{\in Fib}{\longrightarrow} D

    the universally induced morphism from AA into the pullback of B^\hat B along ff is a weak equivalence:

    A B W W C×DB^ B^ (pb) Fib C D. \array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,.
  2. for some factorization of the form

    f:CWC^FibD f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D

    the universally induced morphism from AA into the pullback of D^\hat D along gg is a weak equivalence:

    AWC^×DB. A \overset{\in W}{\longrightarrow} \hat C \underset{D}{\times} B \,.
  3. the above two conditions hold for every such factorization.

(e.g. Goerss-Jardine 96, II (8.14))

Proposition

The conditions in def. are indeed equivalent.

Proof

First assume that the first condition holds, in that

A B W W C×DB^ B^ (pb) Fib C D. \array{ A &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\longrightarrow& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\longrightarrow& D } \,.

Then let

f:CWC^FibD f \colon C \overset{\in W }{\longrightarrow} \hat C \overset{\in Fib}{\longrightarrow} D

be any factorization of ff and consider the pasting diagram (using the pasting law for pullbacks)

A C^×DB B W W (pb) W C×DB^ W C^×DD^ Fib B^ (pb) Fib (pb) Fib C W C^ Fib D, \array{ A &\overset{}{\longrightarrow}& \hat C \underset{D}{\times} B &\longrightarrow& B \\ {}^{\mathllap{\in W}}\downarrow && \downarrow^{\mathrlap{\in W}} &(pb)& \downarrow^{\mathrlap{\in W}} \\ C\underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat C \underset{D}{\times} \hat D &\overset{\in Fib}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in \atop Fib}} &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& \hat C &\underset{\in Fib}{\longrightarrow}& D } \,,

where the inner morphisms are fibrations and weak equivalences, as shown, by the pullback stability of fibrations (prop. ) and then since pullback along fibrations preserves weak equivalences by assumption of right properness (def. ). Hence it follows by two-out-of-three (def. ) that also the comparison morphism AC^×DBA \to \hat C \underset{D}{\times} B is a weak equivalence.

In conclusion, if the homotopy pullback condition is satisfied for one factorization of gg, then it is satisfied for all factorizations of ff. Since the argument is symmetric in ff and gg, this proves the claim.

Remark

In particular, an ordinary pullback square of fibrant objects, one of whose edges is a fibration, is a homotopy pullback square according to def. .

Proposition

Let 𝒞\mathcal{C} be a right proper model category (def. ). Given a diagram in 𝒞\mathcal{C} of the form

A B Fib C W W W D E Fib F \array{ A &\longrightarrow& B &\overset{\in Fib}{\longleftarrow}& C \\ \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} && \downarrow^{\mathrlap{\in W}} \\ D &\longrightarrow& E &\underset{\in Fib}{\longleftarrow}& F }

then the induced morphism on pullbacks is a weak equivalence

A×BCWD×EF. A \underset{B}{\times} C \overset{\in W}{\longrightarrow} D \underset{E}{\times} F \,.
Proof

(The reader should draw the 3-dimensional cube diagram which we describe in words now.)

First consider the universal morphism CE×FCC \to E \underset{F}{\times} C and observe that it is a weak equivalence by right properness (def. ) and two-out-of-three (def. ).

Then consider the universal morphism A×BCA×B(E×FC)A \underset{B}{\times}C \to A \underset{B}{\times}(E \underset{F}{\times}C) and observe that this is also a weak equivalence, since A×BCA \underset{B}{\times} C is the limiting cone of a homotopy pullback square by remark , and since the morphism is the comparison morphism to the pullback of the factorization constructed in the first step.

Now by using the pasting law, then the commutativity of the “left” face of the cube, then the pasting law again, one finds that A×B(E×FC)A×D(DFE×)A \underset{B}{\times} (E \underset{F}{\times} C) \simeq A \underset{D}{\times} (D \underset{E} F{\times}). Again by right properness this implies that A×B(E×FC)D×EFA \underset{B}{\times} (E \underset{F}{\times} C)\to D \underset{E}{\times} F is a weak equivalence.

With this the claim follows by two-out-of-three.

Homotopy pullbacks satisfy the usual abstract properties of pullbacks:

Proposition

Let 𝒞\mathcal{C} be a right proper model category (def. ). If in a commuting square in 𝒞\mathcal{C} one edge is a weak equivalence, then the square is a homotopy pullback square precisely if the opposite edge is a weak equivalence, too.

Proof

Consider a commuting square of the form

A B C W D. \array{ A &\longrightarrow& B \\ \downarrow && \downarrow \\ C &\underset{\in W}{\longrightarrow}& D } \,.

To detect whether this is a homotopy pullback, by def. and prop. , we are to choose any factorization of the right vertical morphism to obtain the pasting composite

A B W C×DB^ W B^ (pb) Fib C W D. \array{ A &\longrightarrow& B \\ \downarrow && \downarrow^{\mathrlap{\in W}} \\ C \underset{D}{\times} \hat B &\overset{\in W}{\longrightarrow}& \hat B \\ \downarrow &(pb)& \downarrow^{\mathrlap{\in Fib}} \\ C &\underset{\in W}{\longrightarrow}& D } \,.

Here the morphism in the middle is a weak equivalence by right properness (def. ). Hence it follows by two-out-of-three that the top left comparison morphism is a weak equivalence (and so the original square is a homotopy pullback) precisely if the top morphism is a weak equivalence.

Proposition

Let 𝒞\mathcal{C} be a right proper model category (def. ).

  1. (pasting law) If in a commuting diagram

    A B C D E F \array{ A &\longrightarrow& B &\longrightarrow& C \\ \downarrow && \downarrow && \downarrow \\ D &\longrightarrow& E &\underset{}{\longrightarrow}& F }

    the square on the right is a homotoy pullback (def. ) then the left square is, too, precisely if the total rectangle is;

  2. in the presence of functorial factorization (def. ) through weak equivalences followed by fibrations:

    every retract of a homotopy pullback square (in the category 𝒞 f \mathcal{C}_f^{\Box} of commuting squares in 𝒞 f\mathcal{C}_f) is itself a homotopy pullback square.

Proof

For the first statement: choose a factorization of CWF^FibFC \overset{\in W}{\to} \hat F \overset{\in Fib}{\to} F, pull it back to a factorization BB^FibEB \to \hat B \overset{\in Fib}{\to} E and assume that BB^B \to \hat B is a weak equivalence, i.e. that the right square is a homotopy pullback. Now use the ordinary pasting law to conclude.

For the second statement: functorially choose a factorization of the two right vertical morphisms of the squares and factor the squares through the pullbacks of the corresponding fibrations along the bottom morphisms, respectively. Now the statement that the squares are homotopy pullbacks is equivalent to their top left vertical morphisms being weak equivalences. Factor these top left morphisms functorially as cofibrations followed by acyclic fibrations. Then the statement that the squares are homotopy pullbacks is equivalent to those top left cofibrations being acyclic. Now the claim follows using that the retract of an acyclic cofibration is an acyclic cofibration (prop. ).

\,

Long fiber sequences

The ordinary fiber, example , of a morphism has the property that taking it twice is always trivial:

*fib(fib(f))fib(f)XfY. \ast \simeq fib(fib(f)) \longrightarrow fib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,.

This is crucially different for the homotopy fiber, def. . Here we discuss how this comes about and what the consequences are.

Proposition

Let 𝒞 f\mathcal{C}_f be a category of fibrant objects of a model category, def. and let f:XYf \colon X \longrightarrow Y be a morphism in its category of pointed objects, def. . Then the homotopy fiber of its homotopy fiber, def. , is isomorphic, in Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}), to the loop space object ΩY\Omega Y of YY (def. , prop. ):

hofib(hofib(XfY))ΩY. hofib(hofib(X \overset{f}{\to}Y)) \simeq \Omega Y \,.
Proof

Assume without restriction that f:XYf \;\colon\; X \longrightarrow Y is already a fibration between fibrant objects in 𝒞\mathcal{C} (otherwise replace and rename). Then its homotopy fiber is its ordinary fiber, sitting in a pullback square

hofib(f) F i X f * Y. \array{ hofib(f) \simeq & F &\overset{i}{\longrightarrow}& X \\ & \downarrow && \downarrow^{\mathrlap{f}} \\ & \ast &\longrightarrow& Y } \,.

In order to compute hofib(hofib(f))hofib(hofib(f)), i.e. hofib(i)hofib(i), we need to replace the fiber inclusion ii by a fibration. Using the factorization lemma for this purpose yields, after a choice of path space object Path(X)Path(X) (def. ), a replacement of the form

F W F× XPath(X) i Fib i˜ X. \array{ F &\overset{\in W}{\longrightarrow}& F \times_X Path(X) \\ &{}_{\mathllap{i}}\searrow& \downarrow^{\mathrlap{\tilde i}}_{\mathrlap{\in Fib}} \\ && X } \,.

Hence hofib(i)hofib(i) is the ordinary fiber of this map:

hofib(hofib(f))F× XPath(X)× X*Ho(𝒞 */). hofib(hofib(f)) \simeq F \times_X Path(X) \times_X \ast \;\;\;\; \in Ho(\mathcal{C}^{\ast/}) \,.

Notice that

F× XPath(X)*× YPath(X) F \times_X Path(X) \; \simeq \; \ast \times_Y Path(X)

because of the pasting law:

F× XPath(X) Path(X) (pb) F i X (pb) f * Y. \array{ F \times_X Path(X) &\longrightarrow& Path(X) \\ \downarrow &(pb)& \downarrow \\ F &\overset{i}{\longrightarrow}& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y } \,.

Hence

hofib(hofib(f))*× YPath(X)× X*. hofib(hofib(f)) \;\simeq\; \ast \times_Y Path(X) \times_X \ast \,.

Now we claim that there is a choice of path space objects Path(X)Path(X) and Path(Y)Path(Y) such that this model for the homotopy fiber (as an object in 𝒞 */\mathcal{C}^{\ast/}) sits in a pullback diagram of the following form:

*× YPath(X)× X* Path(X) WF ΩY Path(Y)× YX (pb) * Y×X. \array{ \ast \times_Y Path(X) \times_X \ast &\longrightarrow& Path(X) \\ \downarrow && \downarrow\mathrlap{\in W \cap F} \\ \Omega Y &\longrightarrow& Path(Y)\times_Y X \\ \downarrow &(pb)& \downarrow \\ \ast &\longrightarrow& Y \times X } \,.

By the pasting law and the pullback stability of acyclic fibrations, this will prove the claim.

To see that the bottom square here is indeed a pullback, check the universal property: A morphism out of any AA into *×Y×XPath(Y)× YX\ast \underset{Y \times X}{\times} Path(Y) \times_Y X is a morphism a:APath(Y)a \colon A \to Path(Y) and a morphism b:AXb \colon A \to X such that p 0(a)=*p_0(a) = \ast, p 1(a)=f(b)p_1(a) = f(b) and b=*b = \ast. Hence it is equivalently just a morphism a:APath(Y)a \colon A \to Path(Y) such that p 0(a)=*p_0(a) = \ast and p 1(a)=*p_1(a) = \ast. This is the defining universal property of ΩY*×YPath(Y)×Y*\Omega Y \coloneqq \ast \underset{Y}{\times} Path(Y) \underset{Y}{\times} \ast.

Now to construct the right vertical morphism in the top square (Quillen 67, page 3.1): Let Path(Y)Path(Y) be any path space object for YY and let Path(X)Path(X) be given by a factorization

(id X,if,id X):XWPath(X)FibX× YPath(Y)× YX (id_X, \; i \circ f, \; id_X) \;\colon\; X \overset{\in W}{\to} Path(X) \overset{\in Fib}{\longrightarrow} X \times_Y Path(Y) \times_Y X

and regarded as a path space object of XX by further comoposing with

(pr 1,pr 3):X× YPath(Y)× YXFibX×X. (pr_1,pr_3)\colon X \times_Y Path(Y) \times_Y X \overset{\in Fib}{\longrightarrow} X \times X \,.

We need to show that Path(X)Path(Y)× YXPath(X)\to Path(Y) \times_Y X is an acyclic fibration.

It is a fibration because X× YPath(Y)× YXPath(Y)× YXX\times_Y Path(Y) \times_Y X \to Path(Y)\times_Y X is a fibration, this being the pullback of the fibration XfYX \overset{f}{\to} Y.

To see that it is also a weak equivalence, first observe that Path(Y)× YXWFibX Path(Y)\times_Y X \overset{\in W \cap Fib}{\longrightarrow} X, this being the pullback of the acyclic fibration of lemma . Hence we have a factorization of the identity as

id X:XWiPath(X)Path(Y)× YXWFibX id_X \;\colon\; X \underoverset{\in W}{i}{\longrightarrow} Path(X) \overset{}{\longrightarrow} Path(Y)\times_Y X \underset{\in W \cap Fib}{\longrightarrow} X

and so finally the claim follows by two-out-of-three (def. ).

Remark

There is a conceptual way to understand prop. as follows: If we draw double arrows to indicate homotopies, then a homotopy fiber (def. ) is depicted by the following filled square:

hofib(f) * X f Y \array{ hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ X &\underset{f}{\longrightarrow}& Y }

just like the ordinary fiber (example ) is given by a plain square

fib(f) * X f Y. \array{ fib(f) &\longrightarrow& \ast \\ \downarrow && \downarrow \\ X &\underset{f}{\longrightarrow}& Y } \,.

One may show that just like the fiber is the universal solution to making such a commuting square (a pullback limit cone def. ), so the homotopy fiber is the universal solution up to homotopy to make such a commuting square up to homotopy – a homotopy pullback homotopy limit cone.

Now just like ordinary pullbacks satisfy the pasting law saying that attaching two pullback squares gives a pullback rectangle, the analogue is true for homotopy pullbacks. This implies that if we take the homotopy fiber of a homotopy fiber, thereby producing this double homotopy pullback square

hofib(g) hofib(f) * g * X f Y \array{ hofib(g) &\longrightarrow& hofib(f) &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow^{\mathrlap{g}} &\swArrow& \downarrow \\ \ast &\longrightarrow& X &\underset{f}{\longrightarrow}& Y }

then the total outer rectangle here is itself a homotopy pullback. But the outer rectangle exhibits the homotopy fiber of the point inclusion, which, via def. and lemma , is the loop space object:

ΩY * * Y. \array{ \Omega Y &\longrightarrow& \ast \\ \downarrow &\swArrow& \downarrow \\ \ast &\longrightarrow& Y } \,.
Proposition

(long homotopy fiber sequences)

Let 𝒞\mathcal{C} be a model category and let f:XYf \colon X \to Y be morphism in the pointed homotopy category Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}) (prop. ). Then:

  1. There is a long sequence to the left in 𝒞 */\mathcal{C}^{\ast/} of the form

    ΩXΩ¯fΩYhofib(f)XfY, \cdots \longrightarrow \Omega X \overset{\overline{\Omega} f}{\longrightarrow} \Omega Y \longrightarrow hofib(f) \longrightarrow X \overset{f}{\longrightarrow} Y \,,

    where each morphism is the homotopy fiber (def. ) of the following one: the homotopy fiber sequence of ff. Here Ω¯f\overline{\Omega}f denotes Ωf\Omega f followed by forming inverses with respect to the group structure on Ω()\Omega(-) from prop. .

    Moreover, for A𝒞 */A\in \mathcal{C}^{\ast/} any object, then there is a long exact sequence

    [A,Ω 2Y] *[A,Ωhofib(f)] *[A,ΩX] *[A,ΩY][A,hofib(f)] *[A,X] *[A,Y] * \cdots \to [A,\Omega^2 Y]_\ast \longrightarrow [A,\Omega hofib(f)]_\ast \longrightarrow [A, \Omega X]_\ast \longrightarrow [A,\Omega Y] \longrightarrow [A,hofib(f)]_\ast \longrightarrow [A,X]_\ast \longrightarrow [A,Y]_\ast

    of pointed sets, where [,] *[-,-]_\ast denotes the pointed set valued hom-functor of example .

  2. Dually, there is a long sequence to the right in 𝒞 */\mathcal{C}^{\ast/} of the form

    XfYhocofib(f)ΣXΣ¯fΣY, X \overset{f}{\longrightarrow} Y \overset{}{\longrightarrow} hocofib(f) \longrightarrow \Sigma X \overset{\overline{\Sigma} f}{\longrightarrow} \Sigma Y \to \cdots \,,

    where each morphism is the homotopy cofiber (def. ) of the previous one: the homotopy cofiber sequence of ff. Moreover, for A𝒞 */A\in \mathcal{C}^{\ast/} any object, then there is a long exact sequence

    [Σ 2X,A] *[Σhocofib(f),A] *[ΣY,A] *[ΣX,A][hocofib(f),A] *[Y,A] *[X,A] * \cdots \to [\Sigma^2 X, A]_\ast \longrightarrow [\Sigma hocofib(f), A]_\ast \longrightarrow [\Sigma Y, A]_\ast \longrightarrow [\Sigma X, A] \longrightarrow [hocofib(f),A]_\ast \longrightarrow [Y,A]_\ast \longrightarrow [X,A]_\ast

    of pointed sets, where [,] *[-,-]_\ast denotes the pointed set valued hom-functor of example .

(Quillen 67, I.3, prop. 4)

Proof

That there are long sequences of this form is the result of combining prop. and prop. .

It only remains to see that it is indeed the morphisms Ω¯f\overline{\Omega} f that appear, as indicated.

In order to see this, it is convenient to adopt the following notation: for f:XYf \colon X \to Y a morphism, then we denote the collection of generalized element of its homotopy fiber as

hofib(f)={(x,f(x)γ 1*)} hofib(f) = \left\{ (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast) \right\}

indicating that these elements are pairs consisting of an element xx of XX and a “path” (an element of the given path space object) from f(x)f(x) to the basepoint.

This way the canonical map hofib(f)Xhofib(f) \to X is (x,f(x)*)x(x, f(x) \rightsquigarrow \ast) \mapsto x. Hence in this notation the homotopy fiber of the homotopy fiber reads

hofib(hofib(f))={((x,f(x)γ 1*),xγ 2*)}. hofib(hofib(f)) = \left\{ ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ) \right\} \,.

This identifies with ΩY\Omega Y by forming the loops

γ 1f(γ 2¯), \gamma_1 \cdot f(\overline{\gamma_2}) \,,

where the overline denotes reversal and the dot denotes concatenation.

Then consider the next homotopy fiber

hofib(hofib(hofib(f)))={(((x,f(x)γ 1*),xγ 2*),(x γ 3 * f(x) f(γ 3) * γ 1 *))}, hofib(hofib(hofib(f))) = \left\{ \left( ( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ), \left( \array{ x && \overset{\gamma_3}{\rightsquigarrow} && \ast \\ f(x) &&\overset{f(\gamma_3)}{\rightsquigarrow}&& \ast \\ & {}_{\mathllap{\gamma_1}}\searrow & \Rightarrow & \swarrow_{\mathllap{}} \\ && \ast } \right) \right) \right\} \,,

where on the right we have a path in hofib(f)hofib(f) from (x,f(x)γ 1*)(x, f(x)\overset{\gamma_1}{\rightsquigarrow} \ast) to the basepoint element. This is a path γ 3\gamma_3 together with a path-of-paths which connects f 1f_1 to f(γ 3)f(\gamma_3).

By the above convention this is identified with the loop in XX which is

γ 2(γ¯ 3). \gamma_2 \cdot (\overline{\gamma}_3) \,.

But the map to hofib(hofib(f))hofib(hofib(f)) sends this data to ((x,f(x)γ 1*),xγ 2*)( (x, f(x) \overset{\gamma_1}{\rightsquigarrow} \ast), x \overset{\gamma_2}{\rightsquigarrow} \ast ), hence to the loop

γ 1f(γ 2¯) f(γ 3)f(γ 2¯) =f(γ 3γ 2¯) =f(γ 2γ¯ 3¯) =f(γ 2γ¯ 3)¯, \begin{aligned} \gamma_1 \cdot f( \overline{\gamma_2} ) & \simeq f(\gamma_3) \cdot f(\overline{\gamma_2}) \\ & = f( \gamma_3 \cdot \overline{\gamma_2} ) \\ & = f ( \overline{\gamma_2 \cdot \overline{\gamma}_3} ) \\ & = \overline{f(\gamma_2 \cdot \overline{\gamma}_3) } \end{aligned} \,,

hence to the reveral of the image under ff of the loop in XX.

Remark

In (Quillen 67, I.3, prop. 3, prop. 4) more is shown than stated in prop. : there the connecting homomorphism ΩYhofib(f)\Omega Y \to hofib(f) is not just shown to exist, but is described in detail via an action of ΩY\Omega Y on hofib(f)hofib(f) in Ho(𝒞)Ho(\mathcal{C}). This takes a good bit more work. For our purposes here, however, it is sufficient to know that such a morphism exists at all, hence that ΩYhofib(hofib(f))\Omega Y \simeq hofib(hofib(f)).

Example

Let 𝒞=(Top cg) Quillen\mathcal{C} = (Top_{cg})_{Quillen} be the classical model structure on topological spaces (compactly generated) from theorem , theorem . Then using the standard pointed topological path space objects Maps(I +,X)Maps(I_+,X) from def. and example as the abstract path space objects in def. , via prop. , this gives that

[*,Ω nX]π n(X) [\ast, \Omega^n X] \simeq \pi_n(X)

is the nnth homotopy group, def. , of XX at its basepoint.

Hence using A=*A = \ast in the first item of prop. , the long exact sequence this gives is of the form

π 3(X)f *π 3(Y)π 2(hofib(f))π 2(X)f *π 2(Y)π 1(hofib(f))π 1(X)f *π 1(Y)*. \cdots \to \pi_3(X) \overset{f_\ast}{\longrightarrow} \pi_3(Y) \longrightarrow \pi_2(hofib(f)) \overset{}{\longrightarrow} \pi_2(X) \overset{-f_\ast}{\longrightarrow} \pi_2(Y) \longrightarrow \pi_1(hofib(f)) \overset{}{\longrightarrow} \pi_1(X) \overset{f_\ast}{\longrightarrow} \pi_1(Y) \overset{}{\longrightarrow} \ast \,.

This is called the long exact sequence of homotopy groups induced by ff.

Remark

As we pass to stable homotopy theory (in Part 1)), the long exact sequences in example become long not just to the left, but also to the right. Given then a tower of fibrations, there is an induced sequence of such long exact sequences of homotopy groups, which organizes into an exact couple. For more on this see at Interlude – Spectral sequences (this remark).

Example

Let again 𝒞=(Top cg) Quillen\mathcal{C} = (Top_{cg})_{Quillen} be the classical model structure on topological spaces (compactly generated) from theorem , theorem , as in example . For ETop cg */E \in Top_{cg}^{\ast/} any pointed topological space and i:AXi \colon A \hookrightarrow X an inclusion of pointed topological spaces, the exactness of the sequence in the second item of prop.

[hocofib(i),E][X,E] *[A,E] * \cdots \to [hocofib(i), E] \longrightarrow [X,E]_\ast \longrightarrow [A,E]_\ast \to \cdots

gives that the functor

[,E] *:(Top CW */) opSet */ [-,E]_\ast \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Set^{\ast/}

behaves like one degree in an additive reduced cohomology theory (def.). The Brown representability theorem (thm.) implies that all additive reduced cohomology theories are degreewise representable this way (prop.).

\,

\infty-Groupoids I): Topological homotopy theory

This section first recalls relevant concepts from actual topology (“point-set topology”) and highlights facts that motivate the axiomatics of model categories below. We prove two technical lemmas (lemma and lemma ) that serve to establish the abstract homotopy theory of topological spaces further below.

Then we discuss how the category Top of topological spaces satisfies the axioms of abstract homotopy theory (model category) theory, def. .

Literature (Hirschhorn 15)

\,

Throughout, let Top denote the category whose objects are topological spaces and whose morphisms are continuous functions between them. Its isomorphisms are the homeomorphisms.

(Further below we restrict attention to the full subcategory of compactly generated topological spaces.)

Universal constructions

To begin with, we recall some basics on universal constructions in Top: limits and colimits of diagrams of topological spaces; exponential objects.

We now discuss limits and colimits (Def. ) in 𝒞=\mathcal{C}= Top. The key for understanding these is the fact that there are initial and final topologies:

Definition

Let {X i=(S i,τ i)Top} iI\{X_i = (S_i,\tau_i) \in Top\}_{i \in I} be a set of topological spaces, and let SSetS \in Set be a bare set. Then

  1. For {Sf iS i} iI\{S \stackrel{f_i}{\to} S_i \}_{i \in I} a set of functions out of SS, the initial topology τ initial({f i} iI)\tau_{initial}(\{f_i\}_{i \in I}) is the topology on SS with the minimum collection of open subsets such that all f i:(S,τ initial({f i} iI))X if_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i are continuous.

  2. For {S if iS} iI\{S_i \stackrel{f_i}{\to} S\}_{i \in I} a set of functions into SS, the final topology τ final({f i} iI)\tau_{final}(\{f_i\}_{i \in I}) is the topology on SS with the maximum collection of open subsets such that all f i:X i(S,τ final({f i} iI))f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I})) are continuous.

Example

For XX a single topological space, and ι S:SU(X)\iota_S \colon S \hookrightarrow U(X) a subset of its underlying set, then the initial topology τ intial(ι S)\tau_{intial}(\iota_S), def. , is the subspace topology, making

ι S:(S,τ initial(ι S))X \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X

a topological subspace inclusion.

Example

Conversely, for p S:U(X)Sp_S \colon U(X) \longrightarrow S an epimorphism, then the final topology τ final(p S)\tau_{final}(p_S) on SS is the quotient topology.

Proposition

Let II be a small category and let X :ITopX_\bullet \colon I \longrightarrow Top be an II-diagram in Top (a functor from II to TopTop), with components denoted X i=(S i,τ i)X_i = (S_i, \tau_i), where S iSetS_i \in Set and τ i\tau_i a topology on S iS_i. Then:

  1. The limit of X X_\bullet exists and is given by the topological space whose underlying set is the limit in Set of the underlying sets in the diagram, and whose topology is the initial topology, def. , for the functions p ip_i which are the limiting cone components:

    lim iIS i p i p j S i S j. \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,.

    Hence

    lim iIX i(lim iIS i,τ initial({p i} iI)) \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)
  2. The colimit of X X_\bullet exists and is the topological space whose underlying set is the colimit in Set of the underlying diagram of sets, and whose topology is the final topology, def. for the component maps ι i\iota_i of the colimiting cocone

    S i S j ι i ι j lim iIS i. \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,.

    Hence

    lim iIX i(lim iIS i,τ final({ι i} iI)) \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right)

(e.g. Bourbaki 71, section I.4)

Proof

The required universal property of (lim iIS i,τ initial({p i} iI))\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) (def. ) is immediate: for

(S,τ) f i f j X i X j \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j }

any cone over the diagram, then by construction there is a unique function of underlying sets Slim iIS iS \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i making the required diagrams commute, and so all that is required is that this unique function is always continuous. But this is precisely what the initial topology ensures.

The case of the colimit is formally dual.

Example

The limit over the empty diagram in TopTop is the point *\ast with its unique topology.

Example

For {X i} iI\{X_i\}_{i \in I} a set of topological spaces, their coproduct iIX iTop\underset{i \in I}{\sqcup} X_i \in Top is their disjoint union.

In particular:

Example

For SSetS \in Set, the SS-indexed coproduct of the point, sS*\underset{s \in S}{\coprod}\ast is the set SS itself equipped with the final topology, hence is the discrete topological space on SS.

Example

For {X i} iI\{X_i\}_{i \in I} a set of topological spaces, their product iIX iTop\underset{i \in I}{\prod} X_i \in Top is the Cartesian product of the underlying sets equipped with the product topology, also called the Tychonoff product.

In the case that SS is a finite set, such as for binary product spaces X×YX \times Y, then a sub-basis for the product topology is given by the Cartesian products of the open subsets of (a basis for) each factor space.

Example

The equalizer of two continuous functions f,g:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the equalizer of the underlying functions of sets

eq(f,g)S XgfS Y eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y

(hence the largets subset of S XS_X on which both functions coincide) and equipped with the subspace topology, example .

Example

The coequalizer of two continuous functions f,g:XYf, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y in TopTop is the coequalizer of the underlying functions of sets

S XgfS Ycoeq(f,g) S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g)

(hence the quotient set by the equivalence relation generated by f(x)g(x)f(x) \sim g(x) for all xXx \in X) and equipped with the quotient topology, example .

Example

For

A g Y f X \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X }

two continuous functions out of the same domain, then the colimit under this diagram is also called the pushout, denoted

A g Y f g *f X X AY.. \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,.

(Here g *fg_\ast f is also called the pushout of ff, or the cobase change of ff along gg.)

This is equivalently the coequalizer of the two morphisms from AA to the coproduct of XX with YY (example ):

AXYX AY. A \stackrel{\longrightarrow}{\longrightarrow} X \sqcup Y \longrightarrow X \sqcup_A Y \,.

If gg is an inclusion, one also writes X fYX \cup_f Y and calls this the attaching space.

By example the pushout/attaching space is the quotient topological space

X AY(XY)/ X \sqcup_A Y \simeq (X\sqcup Y)/\sim

of the disjoint union of XX and YY subject to the equivalence relation which identifies a point in XX with a point in YY if they have the same pre-image in AA.

(graphics from Aguilar-Gitler-Prieto 02)

Notice that the defining universal property of this colimit means that completing the span

A Y X \array{ A &\longrightarrow& Y \\ \downarrow \\ X }

to a commuting square

A Y X Z \array{ A &\longrightarrow& Y \\ \downarrow && \downarrow \\ X &\longrightarrow& Z }

is equivalent to finding a morphism

XAYZ. X \underset{A}{\sqcup} Y \longrightarrow Z \,.
Example

For AXA\hookrightarrow X a topological subspace inclusion, example , then the pushout

A X (po) * X/A \array{ A &\hookrightarrow& X \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& X/A }

is the quotient space or cofiber, denoted X/AX/A.

Example

An important special case of example :

For nn \in \mathbb{N} write

  • D n{x n||x|1} nD^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert \leq 1}\} \hookrightarrow \mathbb{R}^n for the standard topological n-disk (equipped with its subspace topology as a subset of Cartesian space);

  • S n1=D n{x n||x|=1} nS^{n-1} = \partial D^n \coloneqq \{ \vec x\in \mathbb{R}^n | \; {\vert \vec x \vert = 1}\} \hookrightarrow \mathbb{R}^n for its boundary, the standard topological n-sphere.

Notice that S 1=S^{-1} = \emptyset and that S 0=**S^0 = \ast \sqcup \ast.

Let

i n:S n1D n i_n \colon S^{n-1}\longrightarrow D^n

be the canonical inclusion of the standard (n-1)-sphere as the boundary of the standard n-disk (both regarded as topological spaces with their subspace topology as subspaces of the Cartesian space n\mathbb{R}^n).

Then the colimit in Top under the diagram

D ni nS n1i nD n, D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \,,

i.e. the pushout of i ni_n along itself, is the n-sphere S nS^n:

S n1 i n D n i n (po) D n S n. \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,.

(graphics from Ueno-Shiga-Morita 95)

Another kind of colimit that will play a role for certain technical constructions is transfinite composition. First recall

Definition

A partial order is a set SS equipped with a relation \leq such that for all elements a,b,cSa,b,c \in S

1) (reflexivity) aaa \leq a;

2) (transitivity) if aba \leq b and bcb \leq c then aca \leq c;

3) (antisymmetry) if aba\leq b and ba\b \leq a then a=ba = b.

This we may and will equivalently think of as a category with objects the elements of SS and a unique morphism aba \to b precisely if aba\leq b. In particular an order-preserving function between partially ordered sets is equivalently a functor between their corresponding categories.

A bottom element \bot in a partial order is one such that a\bot \leq a for all a. A top element \top is one for wich aa \leq \top.

A partial order is a total order if in addition

4) (totality) either aba\leq b or bab \leq a.

A total order is a well order if in addition

5) (well-foundedness) every non-empty subset has a least element.

An ordinal is the equivalence class of a well-order.

The successor of an ordinal is the class of the well-order with a top element freely adjoined.

A limit ordinal is one that is not a successor.

Example

The finite ordinals are labeled by nn \in \mathbb{N}, corresponding to the well-orders {012n1}\{0 \leq 1 \leq 2 \cdots \leq n-1\}. Here (n+1)(n+1) is the successor of nn. The first non-empty limit ordinal is ω=[(,)]\omega = [(\mathbb{N}, \leq)].

Definition

Let 𝒞\mathcal{C} be a category, and let IMor(𝒞)I \subset Mor(\mathcal{C}) be a class of its morphisms.

For α\alpha an ordinal (regarded as a category), an α\alpha-indexed transfinite sequence of elements in II is a diagram

X :α𝒞 X_\bullet \;\colon\; \alpha \longrightarrow \mathcal{C}

such that

  1. X X_\bullet takes all successor morphisms ββ+1\beta \stackrel{\leq}{\to} \beta + 1 in α\alpha to elements in II

    X β,β+1I X_{\beta,\beta + 1} \in I
  2. X X_\bullet is continuous in that for every nonzero limit ordinal β<α\beta \lt \alpha, X X_\bullet restricted to the full-subdiagram {γ|γβ}\{\gamma \;|\; \gamma \leq \beta\} is a colimiting cocone in 𝒞\mathcal{C} for X X_\bullet restricted to {γ|γ<β}\{\gamma \;|\; \gamma \lt \beta\}.

The corresponding transfinite composition is the induced morphism

X 0X αlimX X_0 \longrightarrow X_\alpha \coloneqq \underset{\longrightarrow}{\lim}X_\bullet

into the colimit of the diagram, schematically:

X 0 X 0,1 X 1 X 1,2 X 2 X α. \array{ X_0 &\stackrel{X_{0,1}}{\to}& X_1 &\stackrel{X_{1,2}}{\to}& X_2 &\to& \cdots \\ & \searrow & \downarrow & \swarrow & \cdots \\ && X_\alpha } \,.

We now turn to the discussion of mapping spaces/exponential objects.

Definition

For XX a topological space and YY a locally compact topological space (in that for every point, every neighbourhood contains a compact neighbourhood), the mapping space

X YTop X^Y \in Top

is the topological space

Accordingly this is called the compact-open topology on the set of functions.

The construction extends to a functor

() ():Top lc op×TopTop. (-)^{(-)} \;\colon\; Top_{lc}^{op} \times Top \longrightarrow Top \,.
Proposition

For XX a topological space and YY a locally compact topological space (in that for each point, each open neighbourhood contains a compact neighbourhood), the topological mapping space X YX^Y from def. is an exponential object, i.e. the functor () Y(-)^Y is right adjoint to the product functor Y×()Y \times (-): there is a natural bijection

Hom Top(Z×Y,X)Hom Top(Z,X Y) Hom_{Top}(Z \times Y, X) \simeq Hom_{Top}(Z, X^Y)

between continuous functions out of any product topological space of YY with any ZTopZ \in Top and continuous functions from ZZ into the mapping space.

A proof is spelled out here (or see e.g. Aguilar-Gitler-Prieto 02, prop. 1.3.1).

Remark

In the context of prop. it is often assumed that YY is also a Hausdorff topological space. But this is not necessary. What assuming Hausdorffness only achieves is that all alternative definitions of “locally compact” become equivalent to the one that is needed for the proposition: for every point, every open neighbourhood contains a compact neighbourhood.

Remark

Proposition fails in general if YY is not locally compact. Therefore the plain category Top of all topological spaces is not a Cartesian closed category.

This is no problem for the construction of the homotopy theory of topological spaces as such, but it becomes a technical nuisance for various constructions that one would like to perform within that homotopy theory. For instance on general pointed topological spaces the smash product is in general not associative.

On the other hand, without changing any of the following discussion one may just pass to a more convenient category of topological spaces such as notably the full subcategory of compactly generated topological spaces Top cgTopTop_{cg} \hookrightarrow Top (def. ) which is Cartesian closed. This we turn to below.

Homotopy

The fundamental concept of homotopy theory is clearly that of homotopy. In the context of topological spaces this is about contiunous deformations of continuous functions parameterized by the standard interval:

Definition

Write

I[0,1] I \coloneqq [0,1] \hookrightarrow \mathbb{R}

for the standard topological interval, a compact connected topological subspace of the real line.

Equipped with the canonical inclusion of its two endpoints

**(δ 0,δ 1)I!* \ast \sqcup \ast \stackrel{(\delta_0,\delta_1)}{\longrightarrow} I \stackrel{\exists !}{\longrightarrow} \ast

this is the standard interval object in Top.

For XTopX \in Top, the product topological space X×IX\times I, example , is called the standard cylinder object over XX. The endpoint inclusions of the interval make it factor the codiagonal on XX

X:XX((id,δ 0),(id,δ 1))X×IX. \nabla_X \;\colon\; X \sqcup X \stackrel{((id,\delta_0),(id,\delta_1))}{\longrightarrow} X \times I \longrightarrow X \,.
Definition

(left homotopy)

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a left homotopy

η:f Lg \eta \colon f \,\Rightarrow_L\, g

is a continuous function

η:X×IY \eta \;\colon\; X \times I \longrightarrow Y

out of the standard cylinder object over XX, def. , such that this fits into a commuting diagram of the form

X (id,δ 0) f X×I η Y (id,δ 1) g X. \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,.

(graphics grabbed from J. Tauber here)

Example

Let XX be a topological space and let x,yXx,y \in X be two of its points, regarded as functions x,y:*Xx,y \colon \ast \longrightarrow X from the point to XX. Then a left homotopy, def. , between these two functions is a commuting diagram of the form

* δ 0 x I η Y δ 1 y *. \array{ \ast \\ {}^{\mathllap{\delta_0}}\downarrow & \searrow^{\mathrlap{x}} \\ I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{\delta_1}}\uparrow & \nearrow_{\mathrlap{y}} \\ \ast } \,.

This is simply a continuous path in XX whose endpoints are xx and yy.

For instance:

Example

Let

const 0:I*δ 0I const_0 \;\colon\; I \longrightarrow \ast \overset{\delta_0}{\longrightarrow} I

be the continuous function from the standard interval I=[0,1]I = [0,1] to itself that is constant on the value 0. Then there is a left homotopy, def. , from the identity function

η:id Iconst 0 \eta \;\colon\; id_I \Rightarrow const_0

given by

η(x,t)x(1t). \eta(x,t) \coloneqq x(1-t) \,.

A key application of the concept of left homotopy is to the definition of homotopy groups:

Definition

For XX a topological space, then its set π 0(X)\pi_0(X) of connected components, also called the 0-th homotopy set, is the set of left homotopy-equivalence classes (def. ) of points x:*Xx \colon \ast \to X, hence the set of path-connected components of XX (example ). By composition this extends to a functor

π 0:TopSet. \pi_0 \colon Top \longrightarrow Set \,.

For nn \in \mathbb{N}, n1n \geq 1 and for x:*Xx \colon \ast \to X any point, then the nnth homotopy group π n(X,x)\pi_n(X,x) of XX at xx is the group

  • whose underlying set is the set of left homotopy-equivalence classes of maps I nXI^n \longrightarrow X that take the boundary of I nI^n to xx and where the left homotopies η\eta are constrained to be constant on the boundary;

  • whose group product operation takes [α:I nX][\alpha \colon I^n \to X] and [β:I nX][\beta \colon I^n \to X] to [αβ][\alpha \cdot \beta] with

αβ:I nI nI n1I n(α,β)X, \alpha \cdot \beta \;\colon\; I^n \stackrel{\simeq}{\longrightarrow} I^n \underset{I^{n-1}}{\sqcup} I^n \stackrel{(\alpha,\beta)}{\longrightarrow} X \,,

where the first map is a homeomorphism from the unit nn-cube to the nn-cube with one side twice the unit length (e.g. (x 1,x 2,x 3,)(2x 1,x 2,x 3,)(x_1, x_2, x_3, \cdots) \mapsto (2 x_1, x_2, x_3, \cdots)).

By composition, this construction extends to a functor

π 1:Top */Grp 1 \pi_{\bullet \geq 1} \;\colon\; Top^{\ast/} \longrightarrow Grp^{\mathbb{N}_{\geq 1}}

from pointed topological spaces to graded groups.

Notice that often one writes the value of this functor on a morphism ff as f *=π (f)f_\ast = \pi_\bullet(f).

Remark

At this point we don’t go further into the abstract reason why def. yields group structure above degree 0, which is that positive dimension spheres are H-cogroup objects. But this is important, for instance in the proof of the Brown representability theorem. See the section Brown representability theorem in Part S.

Definition

(homotopy equivalence)

A continuous function f:XYf \;\colon\; X \longrightarrow Y is called a homotopy equivalence if there exists a continuous function the other way around, g:YXg \;\colon\; Y \longrightarrow X, and left homotopies, def. , from the two composites to the identity:

η 1:fg Lid Y \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y

and

η 2:gf Lid X. \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,.

If here η 2\eta_2 is constant along II, ff is said to exhibit XX as a deformation retract of YY.

Example

For XX a topological space and X×IX \times I its standard cylinder object of def. , then the projection p:X×IXp \colon X \times I \longrightarrow X and the inclusion (id,δ 0):XX×I(id, \delta_0) \colon X \longrightarrow X\times I are homotopy equivalences, def. , and in fact are homotopy inverses to each other:

The composition

X(id,δ 0)X×IpX X \overset{(id,\delta_0)}{\longrightarrow} X\times I \overset{p}{\longrightarrow} X

is immediately the identity on XX (i.e. homotopic to the identity by a trivial homotopy), while the composite

X×IpX(id,δ 0)X×I X \times I \overset{p}{\longrightarrow} X \overset{(id, \delta_0)}{\longrightarrow} X\times I

is homotopic to the identity on X×IX \times I by a homotopy that is pointwise in XX that of example .

Definition

A continuous function f:XYf \colon X \longrightarrow Y is called a weak homotopy equivalence if its image under all the homotopy group functors of def. is an isomorphism, hence if

π 0(f):π 0(X)π 0(X) \pi_0(f) \;\colon\; \pi_0(X) \stackrel{\simeq}{\longrightarrow} \pi_0(X)

and for all xXx \in X and all n1n \geq 1

π n(f):π n(X,x)π n(Y,f(y)). \pi_n(f) \;\colon\; \pi_n(X,x) \stackrel{\simeq}{\longrightarrow} \pi_n(Y,f(y)) \,.
Proposition

Every homotopy equivalence, def. , is a weak homotopy equivalence, def. .

In particular a deformation retraction, def. , is a weak homotopy equivalence.

Proof

First observe that for all XX\in Top the inclusion maps

X(id,δ 0)X×I X \overset{(id,\delta_0)}{\longrightarrow} X \times I

into the standard cylinder object, def. , are weak homotopy equivalences: by postcomposition with the contracting homotopy of the interval from example all homotopy groups of X×IX \times I have representatives that factor through this inclusion.

Then given a general homotopy equivalence, apply the homotopy groups functor to the corresponding homotopy diagrams (where for the moment we notationally suppress the choice of basepoint for readability) to get two commuting diagrams

π (X) π (id,δ 0) π (f)π (g) π (X×I) π (η) π (Y) π (id,δ 1) π (id) π (X),π (Y) π (id,δ 0) π (g)π (f) π (Y×I) π (η) π (X) π (id,δ 1) π (id) π (Y). \array{ \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(f)\circ \pi_\bullet(g)}} \\ \pi_\bullet(X \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(X) } \;\;\;\;\;\;\; \,, \;\;\;\;\;\;\; \array{ \pi_\bullet(Y) \\ {}^{\mathllap{\pi_\bullet(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{\pi_\bullet(g)\circ \pi_\bullet(f)}} \\ \pi_\bullet(Y \times I) &\stackrel{\pi_\bullet(\eta)}{\longrightarrow}& \pi_\bullet(X) \\ {}^{\mathllap{\pi_\bullet(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{\pi_\bullet(id)}} \\ \pi_\bullet(Y) } \,.

By the previous observation, the vertical morphisms here are isomorphisms, and hence these diagrams exhibit π (f)\pi_\bullet(f) as the inverse of π (g)\pi_\bullet(g), hence both as isomorphisms.

Remark

The converse of prop. is not true generally: not every weak homotopy equivalence between topological spaces is a homotopy equivalence. (For an example with full details spelled out see for instance Fritsch, Piccinini: “Cellular Structures in Topology”, p. 289-290).

However, as we will discuss below, it turns out that

  1. every weak homotopy equivalence between CW-complexes is a homotopy equivalence (Whitehead's theorem, cor. );

  2. every topological space is connected by a weak homotopy equivalence to a CW-complex (CW approximation, remark ).

Example

For XTopX\in Top, the projection X×IXX\times I \longrightarrow X from the cylinder object of XX, def. , is a weak homotopy equivalence, def. . This means that the factorization

X:XXX×IX \nabla_X \;\colon\; X \sqcup X \stackrel{}{\hookrightarrow} X\times I \stackrel{\simeq}{\longrightarrow} X

of the codiagonal X\nabla_X in def. , which in general is far from being a monomorphism, may be thought of as factoring it through a monomorphism after replacing XX, up to weak homotopy equivalence, by X×IX\times I.

In fact, further below (prop. ) we see that XXX×IX \sqcup X \to X \times I has better properties than the generic monomorphism has, in particular better homotopy invariant properties: it has the left lifting property against all Serre fibrations EpB E \stackrel{p}{\longrightarrow} B that are also weak homotopy equivalences.

Of course the concept of left homotopy in def. is accompanied by a concept of right homotopy. This we turn to now.

Definition

(path space)

For XX a topological space, its standard topological path space object is the topological path space, hence the mapping space X IX^I, prop. , out of the standard interval II of def. .

Example

The endpoint inclusion into the standard interval, def. , makes the path space X IX^I of def. factor the diagonal on XX through the inclusion of constant paths and the endpoint evaluation of paths:

Δ X:XX I*X IX **IX×X. \Delta_X \;\colon\; X \stackrel{X^{I \to \ast}}{\longrightarrow} X^I \stackrel{X^{\ast \sqcup \ast \to I}}{\longrightarrow} X \times X \,.

This is the formal dual to example . As in that example, below we will see (prop. ) that this factorization has good properties, in that

  1. X I*X^{I \to \ast} is a weak homotopy equivalence;

  2. X **IX^{\ast \sqcup \ast \to I} is a Serre fibration.

So while in general the diagonal Δ X\Delta_X is far from being an epimorphism or even just a Serre fibration, the factorization through the path space object may be thought of as replacing XX, up to weak homotopy equivalence, by its path space, such as to turn its diagonal into a Serre fibration after all.

Definition

(right homotopy)

For f,g:XYf,g\colon X \longrightarrow Y two continuous functions between topological spaces X,YX,Y, then a right homotopy f Rgf \Rightarrow_R g is a continuous function

η:XY I \eta \;\colon\; X \longrightarrow Y^I

into the path space object of XX, def. , such that this fits into a commuting diagram of the form

Y f X δ 0 X η Y I g Y δ 1 Y. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{X^{\delta_0}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^I \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{\delta_1}}} \\ && Y } \,.

Cell complexes

We consider topological spaces that are built consecutively by attaching basic cells.

Definition

Write

I Top{S n1ι nD n} nMor(Top) I_{Top} \coloneqq \left\{ S^{n-1} \stackrel{\iota_n}{\hookrightarrow} D^{n} \right\}_{n \in \mathbb{N}} \; \subset Mor(Top)

for the set of canonical boundary inclusion maps of the standard n-disks, example . This going to be called the set of standard topological generating cofibrations.

Definition

For XTopX \in Top and for nn \in \mathbb{N}, an nn-cell attachment to XX is the pushout (“attaching space”, example ) of a generating cofibration, def.

S n1 ϕ X ι n (po) D n XS n1D n =X ϕD n \array{ S^{n-1} &\stackrel{\phi}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& X \underset{S^{n-1}}{\sqcup} D^n & = X \cup_\phi D^n }

along some continuous function ϕ\phi.

A continuous function f:XYf \colon X \longrightarrow Y is called a topological relative cell complex if it is exhibited by a (possibly infinite) sequence of cell attachments to XX, in that it is a transfinite composition (def. ) of pushouts (example )

iS n i1 X k iι n i (po) iD n i X k+1 \array{ \underset{i}{\coprod} S^{n_i - 1} &\longrightarrow& X_{k} \\ {}^{\mathllap{\underset{i}{\coprod}\iota_{n_i}}}\downarrow &(po)& \downarrow \\ \underset{i}{\coprod} D^{n_i} &\longrightarrow& X_{k+1} }

of coproducts (example ) of generating cofibrations (def. ).

A topological space XX is a cell complex if X\emptyset \longrightarrow X is a relative cell complex.

A relative cell complex is called a finite relative cell complex if it is obtained from a finite number of cell attachments.

A (relative) cell complex is called a (relative) CW-complex if the above transfinite composition is countable

X=X 0 X 1 X 2 f Y=limX \array{ X = X_0 &\longrightarrow& X_1 &\longrightarrow& X_2 &\longrightarrow& \cdots \\ & {}_{\mathllap{f}}\searrow & \downarrow & \swarrow && \cdots \\ && Y = \underset{\longrightarrow}{\lim} X_\bullet }

and if X kX_k is obtained from X k1X_{k-1} by attaching cells precisely only of dimension kk.

Remark

Strictly speaking a relative cell complex, def. , is a function f:XYf\colon X \to Y, together with its cell structure, hence together with the information of the pushout diagrams and the transfinite composition of the pushout maps that exhibit it.

In many applications, however, all that matters is that there is some (relative) cell decomosition, and then one tends to speak loosely and mean by a (relative) cell complex only a (relative) topological space that admits some cell decomposition.

The following lemma , together with lemma below are the only two statements of the entire development here that involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.

Lemma

Assuming the axiom of choice and the law of excluded middle, every compact subspace of a topological cell complex, def. , intersects the interior of a finite number of cells.

(e.g. Hirschhorn 15, section 3.1)

Proof

So let YY be a topological cell complex and CYC \hookrightarrow Y a compact subspace. Define a subset

PY P \subset Y

by choosing one point in the interior of the intersection with CC of each cell of YY that intersects CC.

It is now sufficient to show that PP has no accumulation point. Because, by the compactness of XX, every non-finite subset of CC does have an accumulation point, and hence the lack of such shows that PP is a finite set and hence that CC intersects the interior of finitely many cells of YY.

To that end, let cCc\in C be any point. If cc is a 0-cell in YY, write U c{c}U_c \coloneqq \{c\}. Otherwise write e ce_c for the unique cell of YY that contains cc in its interior. By construction, there is exactly one point of PP in the interior of e ce_c. Hence there is an open neighbourhood cU ce cc \in U_c \subset e_c containing no further points of PP beyond possibly cc itself, if cc happens to be that single point of PP in e ce_c.

It is now sufficient to show that U cU_c may be enlarged to an open subset U˜ c\tilde U_c of YY containing no point of PP, except for possibly cc itself, for that means that cc is not an accumulation point of PP.

To that end, let α c\alpha_c be the ordinal that labels the stage Y α cY_{\alpha_c} of the transfinite composition in the cell complex-presentation of YY at which the cell e ce_c above appears. Let γ\gamma be the ordinal of the full cell complex. Then define the set

T{(β,U)|α cβγ,UopenY β,UY α=U c,UP{,{c}}}, T \coloneqq \left\{ \; (\beta, U) \;|\; \alpha_c \leq \beta \leq \gamma \;\,,\; U \underset{open}{\subset} Y_\beta \;\,,\; U \cap Y_\alpha = U_c \;\,,\; U \cap P \in \{ \emptyset, \{c\} \} \; \right\} \,,

and regard this as a partially ordered set by declaring a partial ordering via

(β 1,U 1)<(β 2,U 2)β 1<β 2,U 2Y β 1=U 1. (\beta_1, U_1) \lt (\beta_2, U_2) \;\;\;\; \Leftrightarrow \;\;\;\; \beta_1 \lt \beta_2 \;\,,\; U_2 \cap Y_{\beta_1} = U_1 \,.

This is set up such that every element (β,U)(\beta, U) of TT with β\beta the maximum value β=γ\beta = \gamma is an extension U˜ c\tilde U_c that we are after.

Observe then that for (β s,U s) sS(\beta_s, U_s)_{s\in S} a chain in (T,<)(T,\lt) (a subset on which the relation <\lt restricts to a total order), it has an upper bound in TT given by the union ( sβ s, sU s)({\cup}_s \beta_s ,\cup_s U_s). Therefore Zorn's lemma applies, saying that (T,<)(T,\lt) contains a maximal element (β max,U max)(\beta_{max}, U_{max}).

Hence it is now sufficient to show that β max=γ\beta_{max} = \gamma. We argue this by showing that assuming β max<γ\beta_{\max}\lt \gamma leads to a contradiction.

So assume β max<γ\beta_{max}\lt \gamma. Then to construct an element of TT that is larger than (β max,U max)(\beta_{max},U_{max}), consider for each cell dd at stage Y β max+1Y_{\beta_{max}+1} its attaching map h d:S n1Y β maxh_d \colon S^{n-1} \to Y_{\beta_{max}} and the corresponding preimage open set h d 1(U max)S n1h_d^{-1}(U_{max})\subset S^{n-1}. Enlarging all these preimages to open subsets of D nD^n (such that their image back in X β max+1X_{\beta_{max}+1} does not contain cc), then (β max,U max)<(β max+1, dU d)(\beta_{max}, U_{max}) \lt (\beta_{max}+1, \cup_d U_d ). This is a contradiction. Hence β max=γ\beta_{max} = \gamma, and we are done.

It is immediate and useful to generalize the concept of topological cell complexes as follows.

Definition

For 𝒞\mathcal{C} any category and for KMor(𝒞)K \subset Mor(\mathcal{C}) any sub-class of its morphisms, a relative KK-cell complexes is a morphism in 𝒞\mathcal{C} which is a transfinite composition (def. ) of pushouts of coproducts of morphsims in KK.

Definition

Write

J Top{D n(id,δ 0)D n×I} nMor(Top) J_{Top} \coloneqq \left\{ D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I \right\}_{n \in \mathbb{N}} \; \subset Mor(Top)

for the set of inclusions of the topological n-disks, def. , into their cylinder objects, def. , along (for definiteness) the left endpoint inclusion.

These inclusions are similar to the standard topological generating cofibrations I TopI_{Top} of def. , but in contrast to these they are “acyclic” (meaning: trivial on homotopy classes of maps from “cycles” given by n-spheres) in that they are weak homotopy equivalences (by prop. ).

Accordingly, J TopJ_{Top} is to be called the set of standard topological generating acyclic cofibrations.

Lemma

For XX a CW-complex (def. ), then its inclusion X(id,δ 0)X×IX \overset{(id, \delta_0)}{\longrightarrow} X\times I into its standard cylinder (def. ) is a J TopJ_{Top}-relative cell complex (def. , def. ).

Proof

First erect a cylinder over all 0-cells

xX 0D 0 X (po) xX 0D 1 Y 1. \array{ \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X \\ \downarrow &(po)& \downarrow \\ \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1 } \,.

Assume then that the cylinder over all nn-cells of XX has been erected using attachment from J TopJ_{Top}. Then the union of any (n+1)(n+1)-cell σ\sigma of XX with the cylinder over its boundary is homeomorphic to D n+1D^{n+1} and is like the cylinder over the cell “with end and interior removed”. Hence via attaching along D n+1D n+1×ID^{n+1} \to D^{n+1}\times I the cylinder over σ\sigma is erected.

Lemma

The maps D nD n×ID^n \hookrightarrow D^n \times I in def. are finite relative cell complexes, def. . In other words, the elements of J TopJ_{Top} are I TopI_{Top}-relative cell complexes.

Proof

There is a homeomorphism

D n = D n (id,δ 0) D n×I D n+1 \array{ D^n & = & D^n \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow \\ D^n \times I &\simeq& D^{n+1} }

such that the map on the right is the inclusion of one hemisphere into the boundary n-sphere of D n+1D^{n+1}. This inclusion is the result of attaching two cells:

S n1 ι n D n ι n (po) D n S n = S n id S n ι n+1 (po) D n+1 id D n+1. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^{n} \\ && \downarrow^{=} \\ S^n &\overset{id}{\longrightarrow}& S^n \\ {}^{\mathllap{\iota_{n+1}}}\downarrow &(po)& \downarrow \\ D^{n+1} &\underset{id}{\longrightarrow}& D^{n+1} } \,.

here the top pushout is the one from example .

Lemma

Every J TopJ_{Top}-relative cell complex (def. , def. ) is a weak homotopy equivalence, def. .

Proof

Let XX^=lim βαX βX \longrightarrow \hat X = \underset{\longleftarrow}{\lim}_{\beta \leq \alpha} X_\beta be a J TopJ_{Top}-relative cell complex.

First observe that with the elements D nD n×ID^n \hookrightarrow D^n \times I of J TopJ_{Top} being homotopy equivalences for all nn \in \mathbb{N} (by example ), each of the stages X βX β+1X_{\beta} \longrightarrow X_{\beta + 1} in the relative cell complex is also a homotopy equivalence. We make this fully explicit:

By definition, such a stage is a pushout of the form

iID n i X β iI(id,δ 0) (po) iID n i×I X β+1. \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow &(po)& \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,.

Then the fact that the projections p n i:D n i×ID n ip_{n_i} \colon D^{n_i} \times I \to D^{n_i} are strict left inverses to the inclusions (id,δ 0)(id, \delta_0) gives a commuting square of the form

iID n i X β iI(id,δ 0) id iID n i×I iIp n i iID n i X β \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta }

and so the universal property of the colimit (pushout) X β+1X_{\beta + 1} gives a factorization of the identity morphism on the right through X β+1X_{\beta + 1}

iID n i X β iI(id,δ 0) iID n i×I X β+1 iIp n i iID n i X β \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ {}^{\mathllap{\underset{i \in I}{\sqcup} (id, \delta_0)} }\downarrow && \downarrow^{} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} \\ {}^{\mathllap{\underset{i \in I}{\sqcup} p_{n_i} }}\downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta }

which exhibits X β+1X βX_{\beta + 1} \to X_\beta as a strict left inverse to X βX β+1X_{\beta} \to X_{\beta + 1}. Hence it is now sufficient to show that this is also a homotopy right inverse.

To that end, let

η n i:D n i×ID n i×I \eta_{n_i} \colon D^{n_i}\times I \longrightarrow D^{n_i} \times I

be the left homotopy that exhibits p n ip_{n_i} as a homotopy right inverse to p n ip_{n_i} by example . For each t[0,1]t \in [0,1] consider the commuting square

iID n i X β iID n i×I X β+1 η n i(,t) id iID n i×I X β+1. \array{ \underset{i \in I}{\sqcup} D^{n_i} &\longrightarrow& X_\beta \\ \downarrow && \downarrow \\ \underset{i \in I}{\sqcup} D^{n_i} \times I && X_{\beta + 1} \\ {}^{\mathllap{\eta_{n_i}(-,t)}}\downarrow && \downarrow^{\mathrlap{id}} \\ \underset{i \in I}{\sqcup} D^{n_i} \times I &\longrightarrow& X_{\beta + 1} } \,.

Regarded as a cocone under the span in the top left, the universal property of the colimit (pushout) X β+1X_{\beta + 1} gives a continuous function

η(,t):X β+1X β+1 \eta(-,t) \;\colon\; X_{\beta + 1} \longrightarrow X_{\beta + 1}

for each t[0,1]t \in [0,1]. For t=0t = 0 this construction reduces to the provious one in that η(,0):X β+1X βX β+1\eta(-,0) \colon X_{\beta +1 } \to X_{\beta} \to X_{\beta + 1} is the composite which we need to homotope to the identity; while η(,1)\eta(-,1) is the identity. Since η(,t)\eta(-,t) is clearly also continuous in tt it constitutes a continuous function

η:X β+1×IX β+1 \eta \;\colon\; X_{\beta + 1}\times I \longrightarrow X_{\beta + 1}

which exhibits the required left homotopy.

So far this shows that each stage X βX β+1X_{\beta} \to X_{\beta+1} in the transfinite composition defining X^\hat X is a homotopy equivalence, hence, by prop. , a weak homotopy equivalence.

This means that all morphisms in the following diagram (notationally suppressing basepoints and showing only the finite stages)

π n(X) π n(X 1) π n(X 2) π n(X 3) lim απ n(X α) \array{ \pi_n(X) &\overset{\simeq}{\longrightarrow}& \pi_n(X_1) &\overset{\simeq}{\longrightarrow}& \pi_n(X_2) &\overset{\simeq}{\longrightarrow}& \pi_n(X_3) &\overset{\simeq}{\longrightarrow}& \cdots \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\simeq}} & \swarrow_{\mathrlap{\simeq}} & \cdots \\ && \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) }

are isomorphisms.

Moreover, lemma gives that every representative and every null homotopy of elements in π n(X^)\pi_n(\hat X) already exists at some finite stage X kX_k. This means that also the universally induced morphism

lim απ n(X α)π n(X^) \underset{\longleftarrow}{\lim}_\alpha \pi_n(X_\alpha) \overset{\simeq}{\longrightarrow} \pi_n(\hat X)

is an isomorphism. Hence the composite π n(X)π n(X^)\pi_n(X) \overset{\simeq}{\longrightarrow} \pi_n(\hat X) is an isomorphism.

Fibrations

Given a relative CC-cell complex ι:XY\iota \colon X \to Y, def. , it is typically interesting to study the extension problem along ff, i.e. to ask which topological spaces EE are such that every continuous function f:XEf\colon X \longrightarrow E has an extension f˜\tilde f along ι\iota

X f E ι f˜ Y. \array{ X &\stackrel{f}{\longrightarrow}& E \\ {}^{\mathllap{\iota}}\downarrow & \nearrow_{\mathrlap{\exists \tilde f}} \\ Y } \,.

If such extensions exists, it means that EE is sufficiently “spread out” with respect to the maps in CC. More generally one considers this extension problem fiberwise, i.e. with both EE and YY (hence also XX) equipped with a map to some base space BB:

Definition

Given a category 𝒞\mathcal{C} and a sub-class CMor(𝒞)C \subset Mor(\mathcal{C}) of its morphisms, then a morphism p:EBp \colon E \longrightarrow B in 𝒞\mathcal{C} is said to have the right lifting property against the morphisms in CC if every commuting diagram in 𝒞\mathcal{C} of the form

X E c p Y B, \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,,

with cCc \in C, has a lift hh, in that it may be completed to a commuting diagram of the form

X E c h p Y B. \array{ X &\longrightarrow& E \\ {}^{\mathllap{c}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &\longrightarrow& B } \,.

We will also say that ff is a CC-injective morphism if it satisfies the right lifting property against CC.

Definition

A continuous function p:EBp \colon E \longrightarrow B is called a Serre fibration if it is a J TopJ_{Top}-injective morphism; i.e. if it has the right lifting property, def. , against all topological generating acylic cofibrations, def. ; hence if for every commuting diagram of continuous functions of the form

D n E (id,δ 0) p D n×I B, \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,,

has a lift hh, in that it may be completed to a commuting diagram of the form

D n E (id,δ 0) h p D n×I B. \array{ D^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow &{}^{\mathllap{h}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,.
Remark

Def. says, in view of the definition of left homotopy, that a Serre fibration pp is a map with the property that given a left homotopy, def. , between two functions into its codomain, and given a lift of one the two functions through pp, then also the homotopy between the two lifts. Therefore the condition on a Serre fibration is also called the homotopy lifting property for maps whose domain is an n-disk.

More generally one may ask functions pp to have such homotopy lifting property for functions with arbitrary domain. These are called Hurewicz fibrations.

Remark

The precise shape of D nD^n and D n×ID^n \times I in def. turns out not to actually matter much for the nature of Serre fibrations. We will eventually find below (prop. ) that what actually matters here is only that the inclusions D nD n×ID^n \hookrightarrow D^n \times I are relative cell complexes (lemma ) and weak homotopy equivalences (prop. ) and that all of these may be generated from them in a suitable way.

But for simple special cases this is readily seen directly, too. Notably we could replace the n-disks in def. with any homeomorphic topological space. A choice important in the comparison to the classical model structure on simplicial sets (below) is to instead take the topological n-simplices Δ n\Delta^n. Hence a Serre fibration is equivalently characterized as having lifts in all diagrams of the form

Δ n E (id,δ 0) p Δ n×I B. \array{ \Delta^n &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ \Delta^n\times I &\longrightarrow& B } \,.

Other deformations of the nn-disks are useful in computations, too. For instance there is a homeomorphism from the nn-disk to its “cylinder with interior and end removed”, formally:

(D n×{0})(D n×I) D n D n×I D n×I \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\simeq& D^n \\ \downarrow && \downarrow \\ D^n \times I &\simeq& D^n\times I }

and hence ff is a Serre fibration equivalently also if it admits lifts in all diagrams of the form

(D n×{0})(D n×I) E (id,δ 0) p D n×I B. \array{ (D^n \times \{0\})\cup (\partial D^n \times I) &\longrightarrow& E \\ {}^{\mathllap{(id,\delta_0)}}\downarrow && \downarrow^{\mathrlap{p}} \\ D^n\times I &\longrightarrow& B } \,.

The following is a general fact about closure of morphisms defined by lifting properties which we prove in generality below as prop. .

Proposition

A Serre fibration, def. has the right lifting property against all retracts (see remark ) of J TopJ_{Top}-relative cell complexes (def. , def. ).

The following statement is foreshadowing the long exact sequences of homotopy groups (below) induced by any fiber sequence, the full version of which we come to below (example ) after having developed more of the abstract homotopy theory.

Proposition

Let f:XYf\colon X \longrightarrow Y be a Serre fibration, def. , let y:*Yy \colon \ast \to Y be any point and write

F yιXfY F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y

for the fiber inclusion over that point. Then for every choice x:*Xx \colon \ast \to X of lift of the point yy through ff, the induced sequence of homotopy groups

π (F y,x)ι *π (X,x)f *π (Y) \pi_{\bullet}(F_y, x) \overset{\iota_\ast}{\longrightarrow} \pi_\bullet(X, x) \overset{f_\ast}{\longrightarrow} \pi_\bullet(Y)

is exact, in that the kernel of f *f_\ast is canonically identified with the image of ι *\iota_\ast:

ker(f *)im(ι *). ker(f_\ast) \simeq im(\iota_\ast) \,.
Proof

It is clear that the image of ι *\iota_\ast is in the kernel of f *f_\ast (every sphere in F yXF_y\hookrightarrow X becomes constant on yy, hence contractible, when sent forward to YY).

For the converse, let [α]π (X,x)[\alpha]\in \pi_{\bullet}(X,x) be represented by some α:S n1X\alpha \colon S^{n-1} \to X. Assume that [α][\alpha] is in the kernel of f *f_\ast. This means equivalently that α\alpha fits into a commuting diagram of the form

S n1 α X f D n κ Y, \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y } \,,

where κ\kappa is the contracting homotopy witnessing that f *[α]=0f_\ast[\alpha] = 0.

Now since xx is a lift of yy, there exists a left homotopy

η:κconst y \eta \;\colon\; \kappa \Rightarrow const_y

as follows:

S n1 α X ι n f D n κ Y (id,δ 1) id D n (id,δ 0) D n×I η Y * y Y \array{ && S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ && D^n &\overset{\kappa}{\longrightarrow}& Y \\ && \downarrow^{\mathrlap{(id,\delta_1)}} && \downarrow^{\mathrlap{id}} \\ D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ \downarrow && && \downarrow \\ \ast && \overset{y}{\longrightarrow} && Y }

(for instance: regard D nD^n as embedded in n\mathbb{R}^n such that 0 n0 \in \mathbb{R}^n is identified with the basepoint on the boundary of D nD^n and set η(v,t)κ(tv)\eta(\vec v,t) \coloneqq \kappa(t \vec v)).

The pasting of the top two squares that have appeared this way is equivalent to the following commuting square

S n1 α X (id,δ 1) f S n1×I (ι n,id) D n×I η Y. \array{ S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{(id,\delta_1)}}\downarrow && && \downarrow^{\mathrlap{f}} \\ S^{n-1} \times I &\overset{(\iota_n, id)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y } \,.

Because ff is a Serre fibration and by lemma and prop. , this has a lift

η˜:S n1×IX. \tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X \,.

Notice that η˜\tilde \eta is a basepoint preserving left homotopy from α=η˜| 1\alpha = \tilde \eta|_1 to some αη˜| 0\alpha' \coloneqq \tilde \eta|_0. Being homotopic, they represent the same element of π n1(X,x)\pi_{n-1}(X,x):

[α]=[α]. [\alpha'] = [\alpha] \,.

But the new representative α\alpha' has the special property that its image in YY is not just trivializable, but trivialized: combining η˜\tilde \eta with the previous diagram shows that it sits in the following commuting diagram

α: S n1 (id,δ 0) S n1×I η˜ X ι n (ι n,id) f D n (id,δ 0) D n×I η Y * y Y. \array{ \alpha' \colon & S^{n-1} &\overset{(id,\delta_0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X \\ & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}} \\ & D^n &\overset{(id,\delta_0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y \\ & \downarrow && && \downarrow \\ & \ast && \overset{y}{\longrightarrow} && Y } \,.

The commutativity of the outer square says that f *αf_\ast \alpha' is constant, hence that α\alpha' is entirely contained in the fiber F yF_y. Said more abstractly, the universal property of fibers gives that α\alpha' factors through F yιXF_y\overset{\iota}{\hookrightarrow} X, hence that [α]=[α][\alpha'] = [\alpha] is in the image of ι *\iota_\ast.

The following lemma , together with lemma above, are the only two statements of the entire development here that crucially involve the concrete particular nature of topological spaces (“point-set topology”), everything beyond that is general abstract homotopy theory.

Lemma

The continuous functions with the right lifting property, def. against the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. , are precisely those which are both weak homotopy equivalences, def. as well as Serre fibrations, def. .

Proof

We break this up into three sub-statements:

A) I TopI_{Top}-injective morphisms are in particular weak homotopy equivalences

Let p:X^Xp \colon \hat X \to X have the right lifting property against I TopI_{Top}

S n1 X^ ι n p D n X \array{ S^{n-1} &\longrightarrow & \hat X \\ {}^{\mathllap{\iota_n}}\downarrow &{}^{\mathllap{\exists}}\nearrow& \downarrow^{\mathrlap{p}} \\ D^n &\longrightarrow& X }

We check that the lifts in these diagrams exhibit π (f)\pi_\bullet(f) as being an isomorphism on all homotopy groups, def. :

For n=0n = 0 the existence of these lifts says that every point of XX is in the image of pp, hence that π 0(X^)π 0(X)\pi_0(\hat X) \to \pi_0(X) is surjective. Let then S 0=**X^S^0 = \ast \coprod \ast \longrightarrow \hat X be a map that hits two connected components, then the existence of the lift says that if they have the same image in π 0(X)\pi_0(X) then they were already the same connected component in X^\hat X. Hence π 0(X^)π 0(X)\pi_0(\hat X)\to \pi_0(X) is also injective and hence is a bijection.

Similarly, for n1n \geq 1, if S nX^S^n \to \hat X represents an element in π n(X^)\pi_n(\hat X) that becomes trivial in π n(X)\pi_n(X), then the existence of the lift says that it already represented the trivial element itself. Hence π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) has trivial kernel and so is injective.

Finally, to see that π n(X^)π n(X)\pi_n(\hat X) \to \pi_n(X) is also surjective, hence bijective, observe that every elements in π n(X)\pi_n(X) is equivalently represented by a commuting diagram of the form

S n1 * X^ D n X = X \array{ S^{n-1} &\longrightarrow& \ast &\longrightarrow& \hat X \\ \downarrow && \downarrow && \downarrow \\ D^n &\longrightarrow& X &=& X }

and so here the lift gives a representative of a preimage in π n(X^)\pi_{n}(\hat X).

B) I TopI_{Top}-injective morphisms are in particular Serre fibrations

By an immediate closure property of lifting problems (we spell this out in generality as prop. , cor. below) an I TopI_{Top}-injective morphism has the right lifting property against all relative cell complexes, and hence, by lemma , it is also a J TopJ_{Top}-injective morphism, hence a Serre fibration.

C) Acyclic Serre fibrations are in particular I TopI_{Top}-injective morphisms

(Hirschhorn 15, section 6).

Let f:XYf\colon X \to Y be a Serre fibration that induces isomorphisms on homotopy groups. In degree 0 this means that ff is an isomorphism on connected components, and this means that there is a lift in every commuting square of the form

S 1= X f D 0=* Y \array{ S^{-1} = \emptyset &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ D^0 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being surjective) and in every commuting square of the form

S 0 X ι 0 f D 1=* Y \array{ S^0 &\longrightarrow& X \\ {}^{\mathllap{\iota_0}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^1 = \ast &\longrightarrow& Y }

(this is π 0(f)\pi_0(f) being injective). Hence we are reduced to showing that for n2n \geq 2 every diagram of the form

S n1 α X ι n f D n κ Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}} \\ D^n &\overset{\kappa}{\longrightarrow}& Y }

has a lift.

To that end, pick a basepoint on S n1S^{n-1} and write xx and yy for its images in XX and YY, respectively

Then the diagram above expresses that f *[α]=0π n1(Y,y)f_\ast[\alpha] = 0 \in \pi_{n-1}(Y,y) and hence by assumption on ff it follows that [α]=0π n1(X,x)[\alpha] = 0 \in \pi_{n-1}(X,x), which in turn mean that there is κ\kappa' making the upper triangle of our lifting problem commute:

S n1 α X ι n κ D n. \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & \nearrow_{\mathrlap{\kappa'}} \\ D^n } \,.

It is now sufficient to show that any such κ\kappa' may be deformed to a ρ\rho' which keeps making this upper triangle commute but also makes the remaining lower triangle commute.

To that end, notice that by the commutativity of the original square, we already have at least this commuting square:

S n1 ι n D n ι n fκ D n κ Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& Y } \,.

This induces the universal map (κ,fκ)(\kappa,f \circ \kappa') from the pushout of its cospan in the top left, which is the n-sphere (see this example):

S n1 ι n D n ι n (po) fκ D n κ S n (κ,fκ) Y. \array{ S^{n-1} &\overset{\iota_n}{\longrightarrow}& D^n \\ {}^{\mathllap{\iota_n}}\downarrow &(po)& \downarrow^{\mathrlap{f \circ \kappa'}} \\ D^n &\underset{\kappa}{\longrightarrow}& S^n \\ && & \searrow^{(\kappa,f \circ \kappa')} \\ && && Y } \,.

This universal morphism represents an element of the nnth homotopy group:

[(κ,fκ)]π n(Y,y). [(\kappa,f \circ \kappa')] \in \pi_n(Y,y) \,.

By assumption that ff is a weak homotopy equivalence, there is a [ρ]π n(X,x)[\rho] \in \pi_{n}(X,x) with

f *[ρ]=[(κ,fκ)] f_\ast [\rho] = [(\kappa,f \circ \kappa')]

hence on representatives there is a lift up to homotopy

X ρ f S n (κ,fκ) Y. \array{ && X \\ &{}^{\mathllap{\rho}}\nearrow_{\mathrlap{\Downarrow}} & \downarrow^{\mathrlap{f}} \\ S^n &\underset{(\kappa,f\circ \kappa')}{\longrightarrow}& Y } \,.

Morever, we may always find ρ\rho of the form (ρ,κ)(\rho', \kappa') for some ρ:D nX\rho' \colon D^n \to X. (“Paste κ\kappa' to the reverse of ρ\rho.”)

Consider then the map

S n(fρ,κ)Y S^n \overset{(f\circ \rho', \kappa)}{\longrightarrow} Y

and observe that this represents the trivial class:

[(fρ,κ)] =[(fρ,fκ)]+[(fκ,κ)] =f *[(ρ,κ)]=[ρ]+[(fκ,κ)] =[(κ,fκ)]+[(fκ,κ)] =0. \begin{aligned} [(f \circ \rho', \kappa)] & = [(f\circ \rho', f\circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = f_\ast \underset{= [\rho]}{\underbrace{[(\rho',\kappa')]}} + [(f\circ \kappa', \kappa)] \\ & = [(\kappa,f \circ \kappa')] + [(f\circ \kappa', \kappa)] \\ & = 0 \end{aligned} \,.

This means equivalently that there is a homotopy

ϕ:fρκ \phi \; \colon \; f\circ \rho' \Rightarrow \kappa

fixing the boundary of the nn-disk.

Hence if we denote homotopy by double arrows, then we have now achieved the following situation

S n1 α X ι n ρ ϕ f D n Y \array{ S^{n-1} &\overset{\alpha}{\longrightarrow}& X \\ {}^{\mathllap{\iota_n}}\downarrow & {}^{\rho'}\nearrow_{\Downarrow^{\phi}} & \downarrow^{\mathrlap{f}} \\ D^n &\longrightarrow& Y }

and it now suffices to show that ϕ\phi may be lifted to a homotopy of just ρ\rho', fixing the boundary, for then the resulting homotopic ρ\rho'' is the desired lift.

To that end, notice that the condition that ϕ:D n×IY\phi \colon D^n \times I \to Y fixes the boundary of the nn-disk means equivalently that it extends to a morphism

S n1S n1×ID n×I(fα,ϕ)Y S^{n-1} \underset{S^{n-1}\times I}{\sqcup} D^n \times I \overset{(f\circ \alpha,\phi)}{\longrightarrow} Y

out of the pushout that identifies in the cylinder over D nD^n all points lying over the boundary. Hence we are reduced to finding a lift in

D n ρ X f S n1S n1×ID n×I (fα,ϕ) Y. \array{ D^n &\overset{\rho'}{\longrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ S^{n-1}\underset{S^{n-1}\times I}{\sqcup} D^n \times I &\overset{(f\circ \alpha,\phi)}{\longrightarrow}& Y } \,.

But inspection of the left map reveals that it is homeomorphic again to D nD n×ID^n \to D^n \times I, and hence the lift does indeed exist.

The classical model structure on topological spaces

Definition

Say that a continuous function, hence a morphism in Top, is

and hence

  • a acyclic classical cofibration if it is a classical cofibration as well as a classical weak equivalence;

  • a acyclic classical fibration if it is a classical fibration as well as a classical weak equivalence.

Write

W cl,Fib cl,Cof clMor(Top) W_{cl},\;Fib_{cl},\;Cof_{cl} \subset Mor(Top)

for the classes of these morphisms, respectively.

We first prove now that the classes of morphisms in def. satisfy the conditions for a model category structure, def. (after some lemmas, this is theorem below). Then we discuss the resulting classical homotopy category (below) and then a few variant model structures whose proof follows immediately along the line of the proof of Top QuillenTop_{Quillen}:

\,

Proposition

The classical weak equivalences, def. , satify two-out-of-three (def. ).

Proof

Since isomorphisms (of homotopy groups) satisfy 2-out-of-3, this property is directly inherited via the very definition of weak homotopy equivalence, def. .

Lemma

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as a classical cofibration followed by an acyclic classical fibration, def. :

f:XCof clX^W clFib clY. f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in W_{cl} \cap Fib_{cl}}{\longrightarrow} Y \,.
Proof

By lemma the set I Top={S n1D n}I_{Top} = \{S^{n-1}\hookrightarrow D^n\} of topological generating cofibrations, def. , has small domains, in the sense of def. (the n-spheres are compact). Hence by the small object argument, prop. , ff factors as an I TopI_{Top}-relative cell complex, def. , hence just a plain relative cell complex, def. , followed by an I TopI_{Top}-injective morphisms, def. :

f:XCof clX^I TopInjY. f \;\colon\; X \stackrel{\in Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in I_{Top} Inj}{\longrightarrow} Y \,.

By lemma the map X^Y\hat X \to Y is both a weak homotopy equivalence as well as a Serre fibration.

Lemma

Every morphism f:XYf\colon X \longrightarrow Y in Top factors as an acyclic classical cofibration followed by a fibration, def. :

f:XW clCof clX^Fib clY. f \;\colon\; X \stackrel{\in W_{cl} \cap Cof_{cl}}{\longrightarrow} \hat X \stackrel{\in Fib_{cl}}{\longrightarrow} Y \,.
Proof

By lemma the set J Top={D nD n×I}J_{Top} = \{D^n \hookrightarrow D^n\times I\} of topological generating acyclic cofibrations, def. , has small domains, in the sense of def. (the n-disks are compact). Hence by the small object argument, prop. , ff factors as an J TopJ_{Top}-relative cell complex, def. , followed by a J topJ_{top}-injective morphisms, def. :

f:XJ TopCellX^J TopInjY. f \;\colon\; X \stackrel{\in J_{Top} Cell}{\longrightarrow} \hat X \stackrel{\in J_{Top} Inj}{\longrightarrow} Y \,.

By definition this makes X^Y\hat X \to Y a Serre fibration, hence a fibration.

By lemma a relative J TopJ_{Top}-cell complex is in particular a relative I TopI_{Top}-cell complex. Hence XX^X \to \hat X is a classical cofibration. By lemma it is also a weak homotopy equivalence, hence a clasical weak equivalence.

Lemma

Every commuting square in Top with the left morphism a classical cofibration and the right morphism a fibration, def.

gCof cl fFib cl \array{ &\longrightarrow& \\ {}^{\mathllap{{g \in} \atop {Cof_{cl}}}}\downarrow && \downarrow^{\mathrlap{{f \in }\atop Fib_{cl}}} \\ &\longrightarrow& }

admits a lift as soon as one of the two is also a classical weak equivalence.

Proof

A) If the fibration ff is also a weak equivalence, then lemma says that it has the right lifting property against the generating cofibrations I TopI_{Top}, and cor. implies the claim.

B) If the cofibration gg on the left is also a weak equivalence, consider any factorization into a relative J TopJ_{Top}-cell complex, def. , def. , followed by a fibration,

g:J TopCellFib cl, g \;\colon\; \stackrel{\in J_{Top} Cell}{\longrightarrow} \stackrel{\in Fib_{cl}}{\longrightarrow} \,,

as in the proof of lemma . By lemma the morphism J TopCell\overset{\in J_{Top} Cell}{\longrightarrow} is a weak homotopy equivalence, and so by two-out-of-three (prop. ) the factorizing fibration is actually an acyclic fibration. By case A), this acyclic fibration has the right lifting property against the cofibration gg itself, and so the retract argument, lemma gives that gg is a retract of a relative J TopJ_{Top}-cell complex. With this, finally cor. implies that ff has the right lifting property against gg.

Finally:

Proposition

The systems (Cof cl,W clFib cl)(Cof_{cl} , W_{cl} \cap Fib_{cl}) and (W clCof cl,Fib cl)(W_{cl} \cap Cof_{cl}, Fib_{cl}) from def. are weak factorization systems.

Proof

Since we have already seen the factorization property (lemma , lemma ) and the lifting properties (lemma ), it only remains to see that the given left/right classes exhaust the class of morphisms with the given lifting property.

For the classical fibrations this is by definition, for the the classical acyclic fibrations this is by lemma .

The remaining statement for Cof clCof_{cl} and W clCof clW_{cl}\cap Cof_{cl} follows from a general argument (here) for cofibrantly generated model categories (def. ), which we spell out:

So let f:XYf \colon X \longrightarrow Y be in (I TopInj)Proj(I_{Top} Inj) Proj, we need to show that then ff is a retract (remark ) of a relative cell complex. To that end, apply the small object argument as in lemma to factor ff as

f:XI TopCellY^I TopInjY. f \;\colon \; X \overset{I_{Top} Cell}{\longrightarrow} \hat Y \overset{\in I_{Top} Inj}{\longrightarrow} Y \,.

It follows that ff has the left lifting property against Y^Y\hat Y \to Y, and hence by the retract argument (lemma ) it is a retract of XICellY^X \overset{I Cell}{\to} \hat Y. This proves the claim for Cof clCof_{cl}.

The analogous argument for W clCof clW_{cl} \cap Cof_{cl}, using the small object argument for J TopJ_{Top}, shows that every f(J TopInj)Projf \in (J_{Top} Inj) Proj is a retract of a J TopJ_{Top}-cell complex. By lemma and lemma a J TopJ_{Top}-cell complex is both an I TopI_{Top}-cell complex and a weak homotopy equivalence. Retracts of the former are cofibrations by definition, and retracts of the latter are still weak homotopy equivalences by lemma . Hence such ff is an acyclic cofibration.

In conclusion, prop. and prop. say that:

Theorem

The classes of morphisms in Mor(Top)Mor(Top) of def. ,

define a model category structure (def. ) Top QuillenTop_{Quillen}, the classical model structure on topological spaces or Serre-Quillen model structure .

In particular

  1. every object in Top QuillenTop_{Quillen} is fibrant;

  2. the cofibrant objects in Top QuillenTop_{Quillen} are the retracts of cell complexes.

Hence in particular the following classical statement is an immediate corollary:

Corollary

(Whitehead theorem)

Every weak homotopy equivalence (def. ) between topological spaces that are homeomorphic to a retract of a cell complex, in particular to a CW-complex (def. ), is a homotopy equivalence (def. ).

Proof

This is the “Whitehead theorem in model categories”, lemma , specialized to Top QuillenTop_{Quillen} via theorem .

In proving theorem we have in fact shown a bit more that stated. Looking back, all the structure of Top QuillenTop_{Quillen} is entirely induced by the set I TopI_{Top} (def. ) of generating cofibrations and the set J TopJ_{Top} (def. ) of generating acyclic cofibrations (whence the terminology). This situation is usefully summarized by the concept of cofibrantly generated model category (Def. ).

This phenomenon will keep recurring and will keep being useful as we construct further model categories, such as the classical model structure on pointed topological spaces (def. ), the projective model structure on topological functors (thm. ), and finally various model structures on spectra which we turn to in the section on stable homotopy theory.

\,

The classical homotopy category

With the classical model structure on topological spaces in hand, we now have good control over the classical homotopy category:

Definition

The Serre-Quillen classical homotopy category is the homotopy category, def. , of the classical model structure on topological spaces Top QuillenTop_{Quillen} from theorem : we write

Ho(Top)Ho(Top Quillen). Ho(Top) \coloneqq Ho(Top_{Quillen}) \,.
Remark

From just theorem , the definition (def. ) gives that

Ho(Top Quillen)(Top Retract(Cell))/ Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim

is the category whose objects are retracts of cell complexes (def. ) and whose morphisms are homotopy classes of continuous functions. But in fact more is true:

Theorem in itself implies that every topological space is weakly equivalent to a retract of a cell complex, def. . But by the existence of CW approximations, this cell complex may even be taken to be a CW complex.

(Better yet, there is Quillen equivalence to the classical model structure on simplicial sets which implies a functorial CW approximation |SingX|W clX{\vert Sing X\vert} \overset{\in W_{cl}}{\longrightarrow} X given by forming the geometric realization of the singular simplicial complex of XX.)

Hence the Serre-Quillen classical homotopy category is also equivalently the category of just the CW-complexes whith homotopy classes of continuous functions between them

Ho(Top Quillen) (Top Retract(Cell))/ (Top CW)/ . \begin{aligned} Ho(Top_{Quillen}) & \simeq (Top_{Retract(Cell)})/_\sim \\ & \simeq (Top_{CW})/_{\sim} \end{aligned} \,.

It follows that the universal property of the homotopy category (theorem )

Ho(Top Quillen)Top[W cl 1] Ho(Top_{Quillen}) \simeq Top[W_{cl}^{-1}]

implies that there is a bijection, up to natural isomorphism, between

  1. functors out of Top CWTop_{CW} which agree on homotopy-equivalent maps;

  2. functors out of all of TopTop which send weak homotopy equivalences to isomorphisms.

This statement in particular serves to show that two different axiomatizations of generalized (Eilenberg-Steenrod) cohomology theories are equivalent to each other. See at Introduction to Stable homotopy theory – S the section generalized cohomology functors (this prop.)

Beware that, by remark , what is not equivalent to Ho(Top Quillen)Ho(Top_{Quillen}) is the category

hTopTop/ hTop \coloneqq Top/_\sim

obtained from all topological spaces with morphisms the homotopy classes of continuous functions. This category is “too large”, the correct homotopy category is just the genuine full subcategory

Ho(Top Quillen)(Top Retract(Cell))/ Top/ =hTop. Ho(Top_{Quillen}) \simeq (Top_{Retract(Cell)})/_\sim \simeq Top/_\sim = \hookrightarrow hTop \,.

Beware also the ambiguity of terminology: “classical homotopy category” some literature refers to hTophTop instead of Ho(Top Quillen)Ho(Top_{Quillen}). However, here we never have any use for hTophTop and will not mention it again.

Proposition

Let XX be a CW-complex, def. . Then the standard topological cylinder of def.

XX(i 0,i 1)X×IX X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} X\times I \longrightarrow X

(obtained by forming the product space with the standard topological interval I=[0,1]I = [0,1]) is indeed a cylinder object in the abstract sense of def. .

Proof

We describe the proof informally. It is immediate how to turn this into a formal proof, but the notation becomes tedious. (One place where it is spelled out completely is Ottina 14, prop. 2.9.)

So let X 0X 1X 2XX_0 \to X_1 \to X_2\to \cdots \to X be a presentation of XX as a CW-complex. Proceed by induction on the cell dimension.

First observe that the cylinder X 0×IX_0 \times I over X 0X_0 is a cell complex: First X 0X_0 itself is a disjoint union of points. Adding a second copy for every point (i.e. attaching along S 1D 0S^{-1}\to D^0) yields X 0X 0X_0 \sqcup X_0, then attaching an inteval between any two corresponding points (along S 0D 1S^0 \to D^1) yields X 0×IX_0 \times I.

So assume that for nn \in \mathbb{N} it has been shown that X n×IX_n \times I has the structure of a CW-complex of dimension (n+1)(n+1). Then for each cell of X n+1X_{n+1}, attach it twice to X n×IX_n \times I, once at X n×{0}X_n \times \{0\}, and once at X n×{1}X_n \times \{1\}.

The result is X n+1X_{n+1} with a hollow cylinder erected over each of its (n+1)(n+1)-cells. Now fill these hollow cylinders (along S n+1D n+1S^{n+1} \to D^{n+1}) to obtain X n+1×IX_{n+1}\times I.

This completes the induction, hence the proof of the CW-structure on X×IX\times I.

The construction also manifestly exhibits the inclusion XX(i 0,i 1)X \sqcup X \overset{(i_0,i_1)}{\longrightarrow} as a relative cell complex.

Finally, it is clear (prop. ) that X×IXX \times I \to X is a weak homotopy equivalence.

Conversely:

Proposition

Let XX be any topological space. Then the standard topological path space object (def. )

XX I(X δ 0,X δ 1)X×X X \longrightarrow X^I \overset{(X^{\delta_0}, X^{\delta_1})}{\longrightarrow} X \times X

(obtained by forming the mapping space, def. , with the standard topological interval I=[0,1]I = [0,1]) is indeed a path space object in the abstract sense of def. .

Proof

To see that const:XX Iconst \colon X\to X^I is a weak homotopy equivalence it is sufficient, by prop. , to exhibit a homotopy equivalence. Let the homotopy inverse be X δ 0:X IXX^{\delta_0} \colon X^I \to X. Then the composite

XconstX IX δ 0X X \overset{const}{\longrightarrow} X^I \overset{X^{\delta_0}}{\longrightarrow} X

is already equal to the identity. The other we round, the rescaling of paths provides the required homotopy

I×X I (t,γ)γ(t()) X I. \array{ I \times X^I &\overset{(t,\gamma)\mapsto \gamma(t\cdot(-))}{\longrightarrow}& X^I } \,.

To see that X IX×XX^I \to X\times X is a fibration, we need to show that every commuting square of the form

D n X I i 0 D n×I X×X \array{ D^n &\longrightarrow& X^I \\ {}^{\mathllap{i_0}}\downarrow && \downarrow^{} \\ D^n \times I &\longrightarrow& X \times X }

has a lift.

Now first use the adjunction (I×())() I(I \times (-))\dashv (-)^I from prop. to rewrite this equivalently as the following commuting square:

D nD n (i 0,i 0) (D n×I)(D n×I) (i 0,i 1) D n×I X. \array{ D^n \sqcup D^n &\overset{(i_0, i_0)}{\longrightarrow}& (D^n \times I) \sqcup (D^n \times I) \\ {}^{\mathllap{(i_0, i_1)}}\downarrow && \downarrow \\ D^n \times I &\longrightarrow& X } \,.

This square is equivalently (example ) a morphism out of the pushout

D n×ID nD n((D n×I)(D n×I))X. D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) \longrightarrow X \,.

By the same reasoning, a lift in the original diagram is now equivalently a lifting in

D n×ID nD n((D n×I)(D n×I)) X (D n×I)×I *. \array{ D^n \times I \underset{D^n \sqcup D^n }{\sqcup} \left((D^n \times I) \sqcup (D^n \times I)\right) &\longrightarrow& X \\ \downarrow && \downarrow \\ (D^n \times I)\times I &\longrightarrow& \ast } \,.

Inspection of the component maps shows that the left vertical morphism here is the inclusion into the square times D nD^n of three of its faces times D nD^n. This is homeomorphic to the inclusion D n+1D n+1×ID^{n+1} \to D^{n+1} \times I (as in remark ). Therefore a lift in this square exsists, and hence a lift in the original square exists.

Model structure on pointed spaces

A pointed object (X,x)(X,x) is of course an object XX equipped with a point x:*Xx \colon \ast \to X, and a morphism of pointed objects (X,x)(Y,y)(X,x) \longrightarrow (Y,y) is a morphism XYX \longrightarrow Y that takes xx to yy. Trivial as this is in itself, it is good to record some basic facts, which we do here.

Passing to pointed objects is also the first step in linearizing classical homotopy theory to stable homotopy theory. In particular, every category of pointed objects has a zero object, hence has zero morphisms. And crucially, if the original category had Cartesian products, then its pointed objects canonically inherit a non-cartesian tensor product: the smash product. These ingredients will be key below in the section on stable homotopy theory.

Definition

Let 𝒞\mathcal{C} be a category and let X𝒞X \in \mathcal{C} be an object.

The slice category 𝒞 /X\mathcal{C}_{/X} is the category whose

  • objects are morphisms A X\array{A \\ \downarrow \\ X} in 𝒞\mathcal{C};

  • morphisms are commuting triangles A B X\array{ A && \longrightarrow && B \\ & {}_{}\searrow && \swarrow \\ && X} in 𝒞\mathcal{C}.

Dually, the coslice category 𝒞 X/\mathcal{C}^{X/} is the category whose

  • objects are morphisms X A\array{X \\ \downarrow \\ A} in 𝒞\mathcal{C};

  • morphisms are commuting triangles X A B\array{ && X \\ & \swarrow && \searrow \\ A && \longrightarrow && B } in 𝒞\mathcal{C}.

There are the canonical forgetful functors

U:𝒞 /X,𝒞 X/𝒞 U \;\colon \; \mathcal{C}_{/X}, \mathcal{C}^{X/} \longrightarrow \mathcal{C}

given by forgetting the morphisms to/from XX.

We here focus on this class of examples:

Definition

For 𝒞\mathcal{C} a category with terminal object *\ast, the coslice category (def. ) 𝒞 */\mathcal{C}^{\ast/} is the corresponding category of pointed objects: its

  • objects are morphisms in 𝒞\mathcal{C} of the form *xX\ast \overset{x}{\to} X (hence an object XX equipped with a choice of point; i.e. a pointed object);

  • morphisms are commuting triangles of the form

    * x y X f Y \array{ && \ast \\ & {}^{\mathllap{x}}\swarrow && \searrow^{\mathrlap{y}} \\ X && \overset{f}{\longrightarrow} && Y }

    (hence morphisms in 𝒞\mathcal{C} which preserve the chosen points).

Remark

In a category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. , the terminal object coincides with the initial object, both are given by *𝒞\ast \in \mathcal{C} itself, pointed in the unique way.

In this situation one says that *\ast is a zero object and that 𝒞 */\mathcal{C}^{\ast/} is a pointed category.

It follows that also all hom-sets Hom 𝒞 */(X,Y)Hom_{\mathcal{C}^{\ast/}}(X,Y) of 𝒞 */\mathcal{C}^{\ast/} are canonically pointed sets, pointed by the zero morphism

0:X!0!Y. 0 \;\colon\; X \overset{\exists !}{\longrightarrow} 0 \overset{\exists !}{\longrightarrow} Y \,.
Definition

Let 𝒞\mathcal{C} be a category with terminal object and finite colimits. Then the forgetful functor U:𝒞 */𝒞U \colon \mathcal{C}^{\ast/} \to \mathcal{C} from its category of pointed objects, def. , has a left adjoint

𝒞 */U() +𝒞 \mathcal{C}^{\ast/} \underoverset {\underset{U}{\longrightarrow}} {\overset{(-)_+}{\longleftarrow}} {\bot} \mathcal{C}

given by forming the disjoint union (coproduct) with a base point (“adjoining a base point”).

Proposition

Let 𝒞\mathcal{C} be a category with all limits and colimits. Then also the category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. , has all limits and colimits.

Moreover:

  1. the limits are the limits of the underlying diagrams in 𝒞\mathcal{C}, with the base point of the limit induced by its universal property in 𝒞\mathcal{C};

  2. the colimits are the limits in 𝒞\mathcal{C} of the diagrams with the basepoint adjoined.

Proof

It is immediate to check the relevant universal property. For details see at slice category – limits and colimits.

Example

Given two pointed objects (X,x)(X,x) and (Y,y)(Y,y), then:

  1. their product in 𝒞 */\mathcal{C}^{\ast/} is simply (X×Y,(x,y))(X\times Y, (x,y));

  2. their coproduct in 𝒞 */\mathcal{C}^{\ast/} has to be computed using the second clause in prop. : since the point *\ast has to be adjoined to the diagram, it is given not by the coproduct in 𝒞\mathcal{C}, but by the pushout in 𝒞\mathcal{C} of the form:

    * x X y (po) Y XY. \array{ \ast &\overset{x}{\longrightarrow}& X \\ {}^{\mathllap{y}}\downarrow &(po)& \downarrow \\ Y &\longrightarrow& X \vee Y } \,.

    This is called the wedge sum operation on pointed objects.

Generally for a set {X i} iI\{X_i\}_{i \in I} in Top */Top^{\ast/}

  1. their product is formed in TopTop as in example , with the new basepoint canonically induced;

  2. their coproduct is formed by the colimit in TopTop over the diagram with a basepoint adjoined, and is called the wedge sum iIX i\vee_{i \in I} X_i.

Example

For XX a CW-complex, def. then for every nn \in \mathbb{N} the quotient (example ) of its nn-skeleton by its (n1)(n-1)-skeleton is the wedge sum, def. , of nn-spheres, one for each nn-cell of XX:

X n/X n1iI nS n. X^n / X^{n-1} \simeq \underset{i \in I_n}{\vee} S^n \,.
Definition

For 𝒞 */\mathcal{C}^{\ast/} a category of pointed objects with finite limits and finite colimits, the smash product is the functor

()():𝒞 */×𝒞 */𝒞 */ (-)\wedge(-) \;\colon\; \mathcal{C}^{\ast/} \times \mathcal{C}^{\ast/} \longrightarrow \mathcal{C}^{\ast/}

given by

XY*XY(X×Y), X \wedge Y \;\coloneqq\; \ast \underset{X\sqcup Y}{\sqcup} (X \times Y) \,,

hence by the pushout in 𝒞\mathcal{C}

XY (id X,y),(x,id Y) X×Y * XY. \array{ X \sqcup Y &\overset{(id_X,y),(x,id_Y) }{\longrightarrow}& X \times Y \\ \downarrow && \downarrow \\ \ast &\longrightarrow& X \wedge Y } \,.

In terms of the wedge sum from def. , this may be written concisely as

XY=X×YXY. X \wedge Y = \frac{X\times Y}{X \vee Y} \,.
Remark

For a general category 𝒞\mathcal{C} in def. , the smash product need not be associative, namely it fails to be associative if the functor ()×Z(-)\times Z does not preserve the quotients involved in the definition.

In particular this may happen for 𝒞=\mathcal{C} = Top.

A sufficient condition for ()×Z(-) \times Z to preserve quotients is that it is a left adjoint functor. This is the case in the smaller subcategory of compactly generated topological spaces, we come to this in prop. below.

These two operations are going to be ubiquituous in stable homotopy theory:

symbolnamecategory theory
XYX \vee Ywedge sumcoproduct in 𝒞 */\mathcal{C}^{\ast/}
XYX \wedge Ysmash producttensor product in 𝒞 */\mathcal{C}^{\ast/}
Example

For X,YTopX, Y \in Top, with X +,Y +Top */X_+,Y_+ \in Top^{\ast/}, def. , then

  • X +Y +(XY) +X_+ \vee Y_+ \simeq (X \sqcup Y)_+;

  • X +Y +(X×Y) +X_+ \wedge Y_+ \simeq (X \times Y)_+.

Proof

By example , X +Y +X_+ \vee Y_+ is given by the colimit in TopTop over the diagram

* X * * Y. \array{ && && \ast \\ && & \swarrow && \searrow \\ X &\,\,& \ast && && \ast &\,\,& Y } \,.

This is clearly X*YX \sqcup \ast \sqcup Y. Then, by definition

X +Y + (X*)×(X*)(X*)(Y*) X×YXY*XY* X×Y*. \begin{aligned} X_+ \wedge Y_+ & \simeq \frac{(X \sqcup \ast) \times (X \sqcup \ast)}{(X\sqcup \ast) \vee (Y \sqcup \ast)} \\ & \simeq \frac{X \times Y \sqcup X \sqcup Y \sqcup \ast}{X \sqcup Y \sqcup \ast} \\ & \simeq X \times Y \sqcup \ast \,. \end{aligned}
Example

Let 𝒞 */=Top */\mathcal{C}^{\ast/} = Top^{\ast/} be pointed topological spaces. Then

I +Top */ I_+ \in Top^{\ast/}

denotes the standard interval object I=[0,1]I = [0,1] from def. , with a djoint basepoint adjoined, def. . Now for XX any pointed topological space, then

X(I +)=(X×I)/({x 0}×I) X \wedge (I_+) = (X \times I)/(\{x_0\} \times I)

is the reduced cylinder over XX: the result of forming the ordinary cyclinder over XX as in def. , and then identifying the interval over the basepoint of XX with the point.

(Generally, any construction in 𝒞\mathcal{C} properly adapted to pointed objects 𝒞 */\mathcal{C}^{\ast/} is called the “reduced” version of the unpointed construction. Notably so for “reduced suspension” which we come to below.)

Just like the ordinary cylinder X×IX\times I receives a canonical injection from the coproduct XXX \sqcup X formed in TopTop, so the reduced cyclinder receives a canonical injection from the coproduct XXX \sqcup X formed in Top */Top^{\ast/}, which is the wedge sum from example :

XXX(I +). X \vee X \longrightarrow X \wedge (I_+) \,.
Example

For (X,x),(Y,y)(X,x),(Y,y) pointed topological spaces with YY a locally compact topological space, then the pointed mapping space is the topological subspace of the mapping space of def.

Maps((Y,y),(X,x)) *(X Y,const x) Maps((Y,y),(X,x))_\ast \hookrightarrow (X^Y, const_x)

on those maps which preserve the basepoints, and pointed by the map constant on the basepoint of XX.

In particular, the standard topological pointed path space object on some pointed XX (the pointed variant of def. ) is the pointed mapping space Maps(I +,X) *Maps(I_+,X)_\ast.

The pointed consequence of prop. then gives that there is a natural bijection

Hom Top */((Z,z)(Y,y),(X,x))Hom Top */((Z,z),Maps((Y,y),(X,x)) *) Hom_{Top^{\ast/}}((Z,z) \wedge (Y,y), (X,x)) \simeq Hom_{Top^{\ast/}}((Z,z), Maps((Y,y),(X,x))_\ast)

between basepoint-preserving continuous functions out of a smash product, def. , with pointed continuous functions of one variable into the pointed mapping space.

Example

Given a morphism f:XYf \colon X \longrightarrow Y in a category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, def. , with finite limits and colimits,

  1. its fiber or kernel is the pullback of the point inclusion

    fib(f) X (pb) f * Y \array{ fib(f) &\longrightarrow& X \\ \downarrow &(pb)& \downarrow^{\mathrlap{f}} \\ \ast &\longrightarrow& Y }
  2. its cofiber or cokernel is the pushout of the point projection

    X f Y (po) * cofib(f). \array{ X &\overset{f}{\longrightarrow}& Y \\ \downarrow &(po)& \downarrow \\ \ast &\longrightarrow& cofib(f) } \,.
Remark

In the situation of example , both the pullback as well as the pushout are equivalently computed in 𝒞\mathcal{C}. For the pullback this is the first clause of prop. . The second clause says that for computing the pushout in 𝒞\mathcal{C}, first the point is to be adjoined to the diagram, and then the colimit over the larger diagram

* X f Y * \array{ \ast \\ & \searrow \\ & & X &\overset{f}{\longrightarrow}& Y \\ & & \downarrow && \\ & & \ast && }

be computed. But one readily checks that in this special case this does not affect the result. (The technical jargon is that the inclusion of the smaller diagram into the larger one in this case happens to be a final functor.)

Proposition

Let 𝒞\mathcal{C} be a model category and let X𝒞X \in \mathcal{C} be an object. Then both the slice category 𝒞 /X\mathcal{C}_{/X} as well as the coslice category 𝒞 X/\mathcal{C}^{X/}, def. , carry model structures themselves – the model structure on a (co-)slice category, where a morphism is a weak equivalence, fibration or cofibration iff its image under the forgetful functor UU is so in 𝒞\mathcal{C}.

In particular the category 𝒞 */\mathcal{C}^{\ast/} of pointed objects, def. , in a model category 𝒞\mathcal{C} becomes itself a model category this way.

The corresponding homotopy category of a model category, def. , we call the pointed homotopy category Ho(𝒞 */)Ho(\mathcal{C}^{\ast/}).

Proof

This is immediate:

By prop. the (co-)slice category has all limits and colimits. By definition of the weak equivalences in the (co-)slice, they satisfy two-out-of-three, def. , because the do in 𝒞\mathcal{C}.

Similarly, the factorization and lifting is all induced by 𝒞\mathcal{C}: Consider the coslice category 𝒞 X/\mathcal{C}^{X/}, the case of the slice category is formally dual; then if

X A f B \array{ && X \\ & \swarrow && \searrow \\ A && \underset{f}{\longrightarrow} && B }

commutes in 𝒞\mathcal{C}, and a factorization of ff exists in 𝒞\mathcal{C}, it uniquely makes this diagram commute

X A C B. \array{ && X \\ & \swarrow &\downarrow& \searrow \\ A &\longrightarrow& C & \longrightarrow& B } \,.

Similarly, if

A C B D \array{ A &\longrightarrow& C \\ \downarrow && \downarrow \\ B &\longrightarrow& D }

is a commuting diagram in 𝒞 X/\mathcal{C}^{X/}, hence a commuting diagram in 𝒞\mathcal{C} as shown, with all objects equipped with compatible morphisms from XX, then inspection shows that any lift in the diagram necessarily respects the maps from XX, too.

Example

For 𝒞\mathcal{C} any model category, with 𝒞 */\mathcal{C}^{\ast/} its pointed model structure according to prop. , then the corresponding homotopy category (def. ) is, by remark , canonically enriched in pointed sets, in that its hom-functor is of the form

[,] *:Ho(𝒞 */) op×Ho(𝒞 */)Set */. [-,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/})^\op \times Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/} \,.
Definition

Write Top Quillen */Top^{\ast/}_{Quillen} for the classical model structure on pointed topological spaces, obtained from the classical model structure on topological spaces Top QuillenTop_{Quillen} (theorem ) via the induced coslice model structure of prop. .

Its homotopy category, def. ,

Ho(Top */)Ho(Top Quillen */) Ho(Top^{\ast/}) \coloneqq Ho(Top_{Quillen}^{\ast/})

we call the classical pointed homotopy category.

Remark

The fibrant objects in the pointed model structure 𝒞 */\mathcal{C}^{\ast/}, prop. , are those that are fibrant as objects of 𝒞\mathcal{C}. But the cofibrant objects in 𝒞 *\mathcal{C}^{\ast} are now those for which the basepoint inclusion is a cofibration in XX.

For 𝒞 */=Top Quillen */\mathcal{C}^{\ast/} = Top^{\ast/}_{Quillen} from def. , then the corresponding cofibrant pointed topological spaces are tyically referred to as spaces with non-degenerate basepoints or . Notice that the point itself is cofibrant in Top QuillenTop_{Quillen}, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.

While the existence of the model structure on Top */Top^{\ast/} is immediate, via prop. , for the discussion of topologically enriched functors (below) it is useful to record that this, too, is a cofibrantly generated model category (def. ), as follows:

Definition

Write

I Top */={S + n1(ι n) +D + n}Mor(Top */) I_{Top^{\ast/}} = \left\{ S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+ \right\} \;\; \subset Mor(Top^{\ast/})

and

J Top */={D + n(id,δ 0) +(D n×I) +}Mor(Top */), J_{Top^{\ast/}} = \left\{ D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+ \right\} \;\;\; \subset Mor(Top^{\ast/}) \,,

respectively, for the sets of morphisms obtained from the classical generating cofibrations, def. , and the classical generating acyclic cofibrations, def. , under adjoining of basepoints (def. ).

Theorem

The sets I Top */I_{Top^{\ast/}} and J Top */J_{Top^{\ast/}} in def. exhibit the classical model structure on pointed topological spaces Top Quillen */Top^{\ast/}_{Quillen} of def. as a cofibrantly generated model category, def. .

(This is also a special case of a general statement about cofibrant generation of coslice model structures, see this proposition.)

Proof

Due to the fact that in J Top */J_{Top^{\ast/}} a basepoint is freely adjoined, lemma goes through verbatim for the pointed case, with J TopJ_{Top} replaced by J Top */J_{Top^{\ast/}}, as do the other two lemmas above that depend on point-set topology, lemma and lemma . With this, the rest of the proof follows by the same general abstract reasoning as above in the proof of theorem .

Model structure on compactly generated spaces

The category Top has the technical inconvenience that mapping spaces X YX^Y (def. ) satisfying the exponential property (prop. ) exist in general only for YY a locally compact topological space, but fail to exist more generally. In other words: Top is not cartesian closed. But cartesian closure is necessary for some purposes of homotopy theory, for instance it ensures that

  1. the smash product (def. ) on pointed topological spaces is associative (prop. below);

  2. there is a concept of topologically enriched functors with values in topological spaces, to which we turn below;

  3. geometric realization of simplicial sets preserves products.

The first two of these are crucial for the development of stable homotopy theory in the next section, the third is a great convenience in computations.

Now, since the homotopy theory of topological spaces only cares about the CW approximation to any topological space (remark ), it is plausible to ask for a full subcategory of Top which still contains all CW-complexes, still has all limits and colimits, still supports a model category structure constructed in the same way as above, but which in addition is cartesian closed, and preferably such that the model structure interacts well with the cartesian closure.

Such a full subcategory exists, the category of compactly generated topological spaces. This we briefly describe now.

Literature (Strickland 09)

\,

Definition

Let XX be a topological space.

A subset AXA \subset X is called compactly closed (or kk-closed) if for every continuous function f:KXf \colon K \longrightarrow X out of a compact Hausdorff space KK, then the preimage f 1(A)f^{-1}(A) is a closed subset of KK.

The space XX is called compactly generated if its closed subsets exhaust (hence coincide with) the kk-closed subsets.

Write

Top cgTop Top_{cg} \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces.

Definition

Write

TopkTop cgTop Top \overset{k}{\longrightarrow} Top_{cg} \hookrightarrow Top

for the functor which sends any topological space X=(S,τ)X = (S,\tau) to the topological space (S,kτ)(S, k \tau) with the same underlying set SS, but with open subsets kτk \tau the collection of all kk-open subsets with respect to τ\tau.

Lemma

Let XTop cgTopX \in Top_{cg} \hookrightarrow Top and let YTopY\in Top. Then continuous functions

XY X \longrightarrow Y

are also continuous when regarded as functions

Xk(Y) X \longrightarrow k(Y)

with kk from def. .

Proof

We need to show that for AXA \subset X a kk-closed subset, then the preimage f 1(A)Xf^{-1}(A) \subset X is closed subset.

Let ϕ:KX\phi \colon K \longrightarrow X be any continuous function out of a compact Hausdorff space KK. Since AA is kk-closed by assumption, we have that (fϕ) 1(A)=ϕ 1(f 1(A))K(f \circ \phi)^{-1}(A) = \phi^{-1}(f^{-1}(A))\subset K is closed in KK. This means that f 1(A)f^{-1}(A) is kk-closed in XX. But by the assumption that XX is compactly generated, it follows that f 1(A)f^{-1}(A) is already closed.

Corollary

For XTop cgX \in Top_{cg} there is a natural bijection

Hom Top(X,Y)Hom Top cg(X,k(Y)). Hom_{Top}(X,Y) \simeq Hom_{Top_{cg}}(X, k(Y)) \,.

This means equivalently that the functor kk (def. ) together with the inclusion from def. forms an pair of adjoint functors

Top cgkTop. Top_{cg} \underoverset {\underset{k}{\longleftarrow}} {\hookrightarrow} {\bot} Top \,.

This in turn means equivalently that Top cgTopTop_{cg} \hookrightarrow Top is a coreflective subcategory with coreflector kk. In particular kk is idemotent in that there are natural homeomorphisms

k(k(X))k(X). k(k(X))\simeq k(X) \,.

Hence colimits in Top cgTop_{cg} exists and are computed as in Top. Also limits in Top cgTop_{cg} exists, these are obtained by computing the limit in Top and then applying the functor kk to the result.

The following is a slight variant of def. , appropriate for the context of Top cgTop_{cg}.

Definition

For X,YTop cgX, Y \in Top_{cg} (def. ) the compactly generated mapping space X YTop cgX^Y \in Top_{cg} is the compactly generated topological space whose underlying set is the set C(Y,X)C(Y,X) of continuous functions f:YXf \colon Y \to X, and for which a subbase for its topology has elements U ϕ(K)U^{\phi(K)}, for UXU \subset X any open subset and ϕ:KY\phi \colon K \to Y a continuous function out of a compact Hausdorff space KK given by

U ϕ(κ){fC(Y,X)|f(ϕ(K))U}. U^{\phi(\kappa)} \coloneqq \left\{ f\in C(Y,X) | f(\phi(K)) \subset U \right\} \,.
Remark

If YY is (compactly generated and) a Hausdorff space, then the topology on the compactly generated mapping space X YX^Y in def. agrees with the compact-open topology of def. . Beware that it is common to say “compact-open topology” also for the topology of the compactly generated mapping space when YY is not Hausdorff. In that case, however, the two definitions in general disagree.

Proposition

The category Top cgTop_{cg} of def. is cartesian closed:

for every XTop cgX \in Top_{cg} then the operation X×()×()×XX\times (-) \times (-)\times X of forming the Cartesian product in Top cgTop_{cg} (which by cor. is kk applied to the usual product topological space) together with the operation () X(-)^X of forming the compactly generated mapping space (def. ) forms a pair of adjoint functors

Top cg() XX×()Top cg. Top_{cg} \underoverset {\underset{(-)^X}{\longrightarrow}} {\overset{X \times (-)}{\longleftarrow}} {\bot} Top_{cg} \,.

For proof see for instance (Strickland 09, prop. 2.12).

Corollary

For X,YTop cg */X, Y \in Top_{cg}^{\ast/}, the operation of forming the pointed mapping space (example ) inside the compactly generated mapping space of def.

Maps(Y,X) *fib(X Yev yX,x) Maps(Y,X)_\ast \coloneqq fib\left( X^Y \overset{ev_y}{\longrightarrow} X \;, x \right)

is left adjoint to the smash product operation on pointed compactly generated topological spaces.

Top cg */Maps(Y,) *Y()Top cg */. Top_{cg}^{\ast/} \underoverset {\underset{Maps(Y,-)_\ast}{\longrightarrow}} {\overset{Y \wedge (-)}{\longleftarrow}} {\bot} Top_{cg}^{\ast/} \,.
Corollary

For II a small category and X :ITop cg */X_\bullet \colon I \to Top^{\ast/}_{cg} a diagram, then the compactly generated mapping space construction from def. preserves limits in its covariant argument and sends colimits in its contravariant argument to limits:

Maps(X,lim iY i) *lim iMaps(X,Y i) * Maps(X,\underset{\longleftarrow}{\lim}_i Y_i)_\ast \;\simeq\; \underset{\longleftarrow}{\lim}_i Maps(X, Y_i)_\ast

and

Maps(lim iX i,Y) *lim iMaps(X i,Y) *. Maps( \underset{\longrightarrow}{\lim}_i X_i, \; Y )_\ast \simeq \underset{\longleftarrow}{\lim}_i Maps( X_i, Y )_\ast \,.
Proof

The first statement is an immediate implication of Maps(X,) *Maps(X,-)_\ast being a right adjoint, according to cor. .

For the second statement, we use that by def. a compactly generated topological space is uniquely determined if one knows all continuous functions out of compact Hausdorff spaces into it. Hence it is sufficient to show that there is a natural isomorphism

Hom Top cg */(K,Maps(lim iX i,Y) *)Hom Top cg */(K,lim iMaps(X i,Y) *) Hom_{Top_{cg}^{\ast/}}\left( K,\; Maps( \underset{\longrightarrow}{\lim}_i X_i, \; Y )_\ast \right) \simeq Hom_{Top^{\ast/}_{cg}}\left( K, \; \underset{\longleftarrow}{\lim}_i Maps( X_i, Y )_\ast \right)

for KK any compact Hausdorff space.

With this, the statement follows by cor. and using that ordinary hom-sets take colimits in the first argument and limits in the second argument to limits:

Hom Top cg */(K,Maps(lim iX i,Y) *) Hom Top cg */(Klim iX i,Y) Hom Top cg */(lim i(KX i),Y) lim i(Hom Top cg */(KX i,Y)) lim iHom Top cg */(K,Maps(X i,Y) *) Hom Top cg */(K,lim iMaps(X i,Y) *). \begin{aligned} Hom_{Top^{\ast/}_{cg}} \left( K, \; Maps(\underset{\longrightarrow}{\lim}_i X_i,\; Y)_\ast \right) & \simeq Hom_{Top^{\ast/}_{cg}} \left( K \wedge \underset{\longrightarrow}{\lim}_i X_i,\; Y \right) \\ & \simeq Hom_{Top^{\ast/}_{cg}} \left( \underset{\longrightarrow}{\lim}_i (K \wedge X_i) ,\; Y \right) \\ & \simeq \underset{\longleftarrow}{\lim}_i \left( Hom_{Top^{\ast/}_{cg}} ( K \wedge X_i, \; Y ) \right) \\ & \simeq \underset{\longleftarrow}{\lim}_i Hom_{Top^{\ast/}_{cg}}( K, \; Maps(X_i,Y)_\ast ) \\ & \simeq Hom_{Top^{\ast/}_{cg}} \left( K,\; \underset{\longleftarrow}{\lim}_i Maps(X_i,Y)_\ast \right) \end{aligned} \,.

Moreover, compact generation fixes the associativity of the smash product (remark ):

Proposition

On pointed (def. ) compactly generated topological spaces (def. ) the smash product (def. )

()():Top cg */×Top cg */Top cg */ (-)\wedge (-) \;\colon\; Top_{cg}^{\ast/} \times Top_{cg}^{\ast/} \longrightarrow Top_{cg}^{\ast/}

is associative and the 0-sphere is a tensor unit for it.

Proof

Since ()×X(-)\times X is a left adjoint by prop. , it presevers colimits and in particular quotient space projections. Therefore with X,Y,ZTop cg */X, Y, Z \in Top_{cg}^{\ast/} then

(XY)Z =X×YX×{y}{x}×Y×Z(XY)×{z}{[x]=[y]}×Z X×Y×ZX×{y}×Z{x}×Y×ZX×Y×{z} X×Y×ZXYZ. \begin{aligned} (X \wedge Y) \wedge Z & = \frac{ \frac{X\times Y}{X \times\{y\} \sqcup \{x\}\times Y} \times Z }{ (X \wedge Y)\times \{z\} \sqcup \{[x] = [y]\} \times Z} \\ & \simeq \frac{\frac{X \times Y \times Z}{X \times \{y\}\times Z \sqcup \{x\}\times Y \times Z}}{ X \times Y \times \{z\} } \\ &\simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z} \end{aligned} \,.

The analogous reasoning applies to yield also X(YZ)X×Y×ZXYZX \wedge (Y\wedge Z) \simeq \frac{X\times Y \times Z}{ X \vee Y \vee Z}.

The second statement follows directly with prop. .

Remark

Corollary together with prop. says that under the smash product the category of pointed compactly generated topological spaces is a closed symmetric monoidal category with tensor unit the 0-sphere.

(Top cg */,,S 0),. (Top_{cg}^{\ast/}, \wedge, S^0) ,.

Notice that by prop. also unpointed compactly generated spaces under Cartesian product form a closed symmetric monoidal category, hence a cartesian closed category

(Top cg,×,*). (Top_{cg}, \times , \ast) \,.

The fact that Top cg */Top_{cg}^{\ast/} is still closed symmetric monoidal but no longer Cartesian exhibits Top cg */Top_{cg}^{\ast/} as being “more linear” than Top cgTop_{cg}. The “full linearization” of Top cgTop_{cg} is the closed symmteric monoidal category of structured spectra under smash product of spectra which we discuss in section 1.

Due to the idempotency kkkk \circ k \simeq k (cor. ) it is useful to know plenty of conditions under which a given topological space is already compactly generated, for then applying kk to it does not change it and one may continue working as in TopTop.

Example

Every CW-complex is compactly generated.

Proof

Since a CW-complex is a Hausdorff space, by prop. and prop. its kk-closed subsets are precisely those whose intersection with every compact subspace is closed.

Since a CW-complex XX is a colimit in Top over attachments of standard n-disks D n iD^{n_i} (its cells), by the characterization of colimits in TopTop (prop.) a subset of XX is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the nn-disks are compact, this implies one direction: if a subset AA of XX intersected with all compact subsets is closed, then AA is closed.

For the converse direction, since a CW-complex is a Hausdorff space and since compact subspaces of Hausdorff spaces are closed, the intersection of a closed subset with a compact subset is closed.

For completeness we record further classes of examples:

(Lewis 78, p. 148)

Recall that by corollary , all colimits of compactly generated spaces are again compactly generated.

Example

The product topological space of a CW-complex with a compact CW-complex, and more generally with a locally compact CW-complex, is compactly generated.

(Hatcher “Topology of cell complexes”, theorem A.6)

More generally:

Proposition

For XX a compactly generated space and YY a locally compact Hausdorff space, then the product topological space X×YX\times Y is compactly generated.

e.g. (Strickland 09, prop. 26)

Finally we check that the concept of homotopy and homotopy groups does not change under passing to compactly generated spaces:

Proposition

For every topological space XX, the canonical function k(X)Xk(X) \longrightarrow X (the adjunction unit) is a weak homotopy equivalence.

Proof

By example , example and lemma , continuous functions S nk(X)S^n \to k(X) and their left homotopies S n×Ik(X)S^n \times I \to k(X) are in bijection with functions S nXS^n \to X and their homotopies S n×IXS^n \times I \to X.

Theorem

The restriction of the model category structure on Top QuillenTop_{Quillen} from theorem along the inclusion Top cgTopTop_{cg} \hookrightarrow Top of def. is still a model category structure, which is cofibrantly generated by the same sets I TopI_{Top} (def. ) and J TopJ_{Top} (def. ) The coreflection of cor. is a Quillen equivalence (def. )

(Top cg) QuillenkTop Quillen. (Top_{cg})_{Quillen} \underoverset {\underset{k}{\longleftarrow}} {\hookrightarrow} {\bot} Top_{Quillen} \,.
Proof

By example , the sets I TopI_{Top} and J TopJ_{Top} are indeed in Mor(Top cg)Mor(Top_{cg}). By example all arguments above about left homotopies between maps out of these basic cells go through verbatim in Top cgTop_{cg}. Hence the three technical lemmas above depending on actual point-set topology, topology, lemma , lemma and lemma , go through verbatim as before. Accordingly, since the remainder of the proof of theorem of Top QuillenTop_{Quillen} follows by general abstract arguments from these, it also still goes through verbatim for (Top cg) Quillen(Top_{cg})_{Quillen} (repeatedly use the small object argument and the retract argument to establish the two weak factorization systems).

Hence the (acyclic) cofibrations in (Top cg) Quillen(Top_{cg})_{Quillen} are identified with those in Top QuillenTop_{Quillen}, and so the inclusion is a part of a Quillen adjunction (def. ). To see that this is a Quillen equivalence (def. ), it is sufficient to check that for XX a compactly generated space then a continuous function f:XYf \colon X \longrightarrow Y is a weak homotopy equivalence (def. ) precisely if the adjunct f˜:Xk(Y)\tilde f \colon X \to k(Y) is a weak homotopy equivalence. But, by lemma , f˜\tilde f is the same function as ff, just considered with different codomain. Hence the result follows with prop. .

\,

Compactly generated weakly Hausdorff topological spaces

While the inclusion Top cgTopTop_{cg} \hookrightarrow Top of def. does satisfy the requirement that it gives a cartesian closed category with all limits and colimits and containing all CW-complexes, one may ask for yet smaller subcategories that still share all these properties but potentially exhibit further convenient properties still.

A popular choice introduced in (McCord 69) is to add the further restriction to topopological spaces which are not only compactly generated but also weakly Hausdorff. This was motivated from (Steenrod 67) where compactly generated Hausdorff spaces were used by the observation ((McCord 69, section 2)) that Hausdorffness is not preserved my many colimit operations, notably not by forming quotient spaces.

On the other hand, in above we wouldn’t have imposed Hausdorffness in the first place. More intrinsic advantages of Top cgwHTop_{cgwH} over Top cgTop_{cg} are the following:

  • every pushout of a morphism in Top cgwHTopTop_{cgwH} \hookrightarrow Top along a closed subspace inclusion in TopTop is again in Top cgwHTop_{cgwH}

  • in Top cgwHTop_{cgwH} quotient spaces are not only preserved by cartesian products (as is the case for all compactly generated spaces due to X×()X\times (-) being a left adjoint, according to cor. ) but by all pullbacks

  • in Top cgwHTop_{cgwH} the regular monomorphisms are the closed subspace inclusions

We will not need this here or in the following sections, but we briefly mention it for completenes:

Definition

A topological space XX is called weakly Hausdorff if for every continuous function

f:KX f \;\colon\; K \longrightarrow X

out of a compact Hausdorff space KK, its image f(K)Xf(K) \subset X is a closed subset of XX.

Proposition

Every Hausdorff space is a weakly Hausdorff space, def. .

Proposition

For XX a weakly Hausdorff topological space, def. , then a subset AXA \subset X is kk-closed, def. , precisely if for every subset KXK \subset X that is compact Hausdorff with respect to the subspace topology, then the intersection KAK \cap A is a closed subset of XX.

e.g. (Strickland 09, lemma 1.4 (c))

Topological enrichment

So far the classical model structure on topological spaces which we established in theorem , as well as the projective model structures on topologically enriched functors induced from it in theorem , concern the hom-sets, but not the hom-spaces (def. ), i.e. the model structure so far has not been related to the topology on hom-spaces. The following statements say that in fact the model structure and the enrichment by topology on the hom-spaces are compatible in a suitable sense: we have an “enriched model category”. This implies in particular that the product/hom-adjunctions are Quillen adjunctions, which is crucial for a decent discusson of the derived functors of the suspension/looping adjunction below.

Definition

Let i 1:X 1Y 1i_1 \colon X_1 \to Y_1 and i 2:X 2Y 2i_2 \colon X_2 \to Y_2 be morphisms in Top cgTop_{cg}, def. . Their pushout product

i 1i 2((id,i 2),(i 1,id)) i_1\Box i_2 \coloneqq ((id, i_2), (i_1,id))

is the universal morphism in the following diagram

X 1×X 2 (i 1,id) (id,i 2) Y 1×X 2 (po) X 1×Y 2 (Y 1×X 2)X 1×X 2(X 1×Y 2) ((id,i 2),(i 1,id)) Y 1×Y 2 \array{ && X_1 \times X_2 \\ & {}^{\mathllap{(i_1,id)}}\swarrow && \searrow^{\mathrlap{(id,i_2)}} \\ Y_1 \times X_2 && (po) && X_1 \times Y_2 \\ & {}_{\mathllap{}}\searrow && \swarrow \\ && (Y_1 \times X_2) \underset{X_1 \times X_2}{\sqcup} (X_1 \times Y_2) \\ && \downarrow^{\mathrlap{((id, i_2), (i_1,id))}} \\ && Y_1 \times Y_2 }
Example

If i 1:X 1Y 1i_1 \colon X_1 \hookrightarrow Y_1 and i 2:X 2Y 2i_2 \colon X_2 \hookrightarrow Y_2 are inclusions, then their pushout product i 1i 2i_1 \Box i_2 from def. is the inclusion

(X 1×Y 2Y 1×X 2)Y 1×Y 2. \left( X_1 \times Y_2 \;\cup\; Y_1 \times X_2 \right) \hookrightarrow Y_1 \times Y_2 \,.

For instance

({0}I)({0}I) \left( \{0\} \hookrightarrow I \right) \Box \left( \{0\} \hookrightarrow I \right)

is the inclusion of two adjacent edges of a square into the square.

Example

The pushout product with an initial morphism is just the ordinary Cartesian product functor

(X)()X×(), (\emptyset \to X) \Box (-) \simeq X \times (-) \,,

i.e.

(X)(AfB)(X×AX×fX×B). (\emptyset \to X) \Box (A \overset{f}{\to} B) \simeq (X\times A \overset{X \times f}{\longrightarrow} X \times B ) \,.
Proof

The product topological space with the empty space is the empty space, hence the map ×A(id,f)×B\emptyset \times A \overset{(id,f)}{\longrightarrow} \emptyset \times B is an isomorphism, and so the pushout in the pushout product is X×AX \times A. From this one reads off the universal map in question to be X×fX \times f:

×A X×A (po) ×B X×A ((id,f),!) X×B. \array{ && \emptyset \times A \\ & {}^{\mathllap{}}\swarrow && \searrow^{\mathrlap{\simeq}} \\ X \times A && (po) && \emptyset \times B \\ & {}_{\mathllap{\simeq}}\searrow && \swarrow \\ && X \times A \\ && \downarrow^{\mathrlap{((id, f), \exists !)}} \\ && X \times B } \,.
Example

With

I Top:{S n1i nD n}andJ Top:{D nj nD n×I} I_{Top} \colon \{ S^{n-1} \overset{i_n}{\hookrightarrow} D^n\} \;\;\; and \;\;\; J_{Top} \colon \{ D^n \overset{j_n}{\hookrightarrow} D^n \times I\}

the generating cofibrations (def. ) and generating acyclic cofibrations (def. ) of (Top cg) Quillen(Top_{cg})_{Quillen} (theorem ), then their pushout-products (def. ) are

i n 1i n 2 i n 1+n 2 i n 1j n 2 j n 1+n 2. \begin{aligned} i_{n_1} \Box i_{n_2} & \simeq i_{n_1 + n_2} \\ i_{n_1} \Box j_{n_2} & \simeq j_{n_1 + n_2} \end{aligned} \,.
Proof

To see this, it is profitable to model n-disks and n-spheres, up to homeomorphism, as nn-cubes D n[0,1] n nD^\n \simeq [0,1]^n \subset \mathbb{R}^n and their boundaries S n1[0,1] nS^{n-1} \simeq \partial [0,1]^n . For the idea of the proof, consider the situation in low dimensions, where one readily sees pictorially that

i 1i 1:(=||) i_1 \Box i_1 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert\vert\;\;\right) \hookrightarrow \Box

and

i 1j 0:(=|). i_1 \Box j_0 \;\colon\; \left(\;\; = \;\;\cup\;\; \vert \;\; \right) \hookrightarrow \Box \,.

Generally, D nD^n may be represented as the space of nn-tuples of elements in [0,1][0,1], and S nS^n as the suspace of tuples for which at least one of the coordinates is equal to 0 or to 1.

Accordingly, S n 1×D n 2D n 1+n 2S^{n_1} \times D^{n_2} \hookrightarrow D^{n_1 + n_2} is the subspace of (n 1+n 2)(n_1+n_2)-tuples, such that at least one of the first n 1n_1 coordinates is equal to 0 or 1, while D n 1×S n 2D n 1+n 2D^{n_1} \times S^{n_2} \hookrightarrow D^{n_1+ n_2} is the subspace of (n 1+n 2)(n_1 + n_2)-tuples such that east least one of the last n 2n_2 coordinates is equal to 0 or to 1. Therefore

S n 1×D n 2D n 1×S n 2S n 1+n 2. S^{n_1} \times D^{n_2} \cup D^{n_1} \times S^{n_2} \simeq S^{n_1 + n_2} \,.

And of course it is clear that D n 1×D n 2D n 1+n 2D^{n_1} \times D^{n_2} \simeq D^{n_1 + n_2}. This shows the first case.

For the second, use that S n 1×D n 2×IS^{n_1} \times D^{n_2} \times I is contractible to S n 1×D n 2S^{n_1} \times D^{n_2} in D n 1×D n 2×ID^{n_1} \times D^{n_2} \times I, and that S n 1×D n 2S^{n_1} \times D^{n_2} is a subspace of D n 1×D n 2D^{n_1} \times D^{n_2}.

Definition

Let i:ABi \colon A \to B and p:XYp \colon X \to Y be two morphisms in Top cgTop_{cg}, def. . Their pullback powering is

p i(p B,X i) p^{\Box i} \coloneqq (p^B, X^i)

being the universal morphism in

X B (p B,X i) Y B×Y AX A Y B (pb) X A Y i p A Y A \array{ && X^B \\ && \downarrow^{\mathrlap{(p^B, X^i)}} \\ && Y^B \underset{Y^A}{\times} X^A \\ & \swarrow && \searrow \\ Y^B && (pb) && X^A \\ & {}_{\mathllap{Y^i}}\searrow && \swarrow_{\mathrlap{p^A}} \\ && Y^A }
Proposition

Let i 1,i 2,pi_1, i_2 , p be three morphisms in Top cgTop_{cg}, def. . Then for their pushout-products (def. ) and pullback-powerings (def. ) the following lifting properties are equivalent (“Joyal-Tierney calculus”):

i 1i 2 has LLP against p i 1 has LLP against p i 2 i 2 has LLP against p i 1. \array{ & i_1 \Box i_2 & \text{has LLP against} & p \\ \Leftrightarrow & i_1 & \text{has LLP against} & p^{\Box i_2} \\ \Leftrightarrow & i_2 & \text{has LLP against} & p^{\Box i_1} } \,.
Proof

We claim that by the cartesian closure of Top cgTop_{cg}, and carefully collecting terms, one finds a natural bijection between commuting squares and their lifts as follows:

Q f X B i 1 p i 2 P (g 1,g 2) Y B×Y AX AQ×BQ×AP×A (f˜,g˜ 2) X i 1i 2 p P×B g˜ 1 Y, \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \;\;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \,,

where the tilde denotes product/hom-adjuncts, for instance

Pg 1Y BP×Bg˜ 1Y \frac{ P \overset{g_1}{\longrightarrow} Y^B }{ P \times B \overset{\tilde g_1}{\longrightarrow} Y }

etc.

To see this in more detail, observe that both squares above each represent two squares from the two components into the fiber product and out of the pushout, respectively, as well as one more square exhibiting the compatibility condition on these components:

Q f X B i 1 p i 2 P (g 1,g 2) Y B×Y AX A {Q f X B i 1 p B P g 1 Y B,Q f X B i 1 X i 2 P g 1 X A,P g 2 X A g 1 p A Y B Y i 2 Y A} {Q×B f˜ X (i 1,id) p P×B g˜ 2 Y,Q×A (id,i 2) Q×B (i 1,id) f˜ P×A g˜ 2 X,P×A g˜ 2 X (id,i 2) p P×B g˜ 1 Y} Q×BQ×AP×A (f˜,g˜ 2) X i 1i 2 p P×B g˜ 1 Y. \begin{aligned} & \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^{\Box i_2}}} \\ P &\underset{(g_1,g_2)}{\longrightarrow}& Y^B \underset{Y^A}{\times} X^A } \\ \simeq & \;\;\;\; \left\{ \;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{p^B}} \\ P &\underset{g_1}{\longrightarrow}& Y^B } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q &\overset{f}{\longrightarrow}& X^B \\ {}^{\mathllap{i_1}}\downarrow && \downarrow^{\mathrlap{X^{i_2}}} \\ P &\underset{g_1}{\longrightarrow}& X^A } \;\;\;\;\; \,, \;\;\;\;\; \array{ P &\overset{g_2}{\longrightarrow}& X^A \\ {}^{\mathllap{g_1}}\downarrow && \downarrow^{\mathrlap{p^A}} \\ Y^B &\underset{Y^{i_2}}{\longrightarrow}& Y^A } \;\;\;\;\; \right\} \\ \leftrightarrow & \;\;\;\; \left\{ \;\;\;\;\; \array{ Q \times B &\overset{\tilde f}{\longrightarrow}& X \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_2}{\longrightarrow}& Y } \;\;\;\;\; \,, \;\;\;\;\; \array{ Q \times A &\overset{(id,i_2)}{\longrightarrow}& Q \times B \\ {}^{\mathllap{(i_1,id)}}\downarrow && \downarrow^{\mathrlap{\tilde f}} \\ P \times A &\underset{\tilde g_2}{\longrightarrow}& X } \;\;\;\;\; \,, \;\;\;\;\; \array{ P \times A &\overset{\tilde g_2}{\longrightarrow}& X \\ {}^{\mathllap{(id,i_2)}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B &\underset{\tilde g_1}{\longrightarrow}& Y } \;\;\;\;\; \right\} \\ \simeq & \;\;\;\; \array{ Q \times B \underset{Q \times A}{\sqcup} P \times A &\overset{(\tilde f, \tilde g_2)}{\longrightarrow}& X \\ {}^{\mathllap{i_1 \Box i_2}}\downarrow && \downarrow^{\mathrlap{p}} \\ P \times B & \underset{\tilde g_1}{\longrightarrow} & Y } \end{aligned} \,.
Proposition

The pushout-product in Top cgTop_{cg} (def. ) of two classical cofibrations is a classical cofibration:

Cof clCof clCof cl. Cof_{cl} \Box Cof_{cl} \subset Cof_{cl} \,.

If one of them is acyclic, then so is the pushout-product:

Cof cl(W clCof cl)W clCof cl. Cof_{cl} \Box (W_{cl} \cap Cof_{cl}) \subset W_{cl}\cap Cof_{cl} \,.
Proof

Regarding the first point:

By example we have

I TopI TopI Top I_{Top} \Box I_{Top} \subset I_{Top}

Hence

I TopI Top has LLP against W clFib cl I Top has LLP against (W clFib cl) I Top Cof cl has LLP against (W clFib cl) I Top I TopCof cl has LLP against W clFib cl I Top has LLP against (W clFib cl) Cof cl Cof cl has LLP against (W clFib cl) Cof cl Cof clCof cl has LLP against W clFib cl, \array{ & I_{Top} \Box I_{Top} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{\Box I_{Top}} \\ \Leftrightarrow & I_{Top} \Box Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} \\ \Leftrightarrow & I_{Top} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Rightarrow & Cof_{cl} & \text{has LLP against} & (W_{cl} \cap Fib_{cl})^{Cof_{cl}} \\ \Leftrightarrow & Cof_{cl} \Box \Cof_{cl} & \text{has LLP against} & W_{cl} \cap Fib_{cl} } \,,

where all logical equivalences used are those of prop. and where all implications appearing are by the closure property of lifting problems, prop. .

Regarding the second point: By example we moreover have

I TopJ TopJ Top I_{Top} \Box J_{Top} \subset J_{Top}

and the conclusion follows by the same kind of reasoning.

Remark

In model category theory the property in proposition is referred to as saying that the model category (Top cg) Quillen(Top_{cg})_{Quillen} from theorem

  1. is a monoidal model category with respect to the Cartesian product on Top cgTop_{cg};

  2. is an enriched model category, over itself.

A key point of what this entails is the following:

Proposition

For X(Top cg) QuillenX \in (Top_{cg})_{Quillen} cofibrant (a retract of a cell complex) then the product-hom-adjunction for YY (prop. ) is a Quillen adjunction

(Top cg) Quillen() XX×()(Top cg) Quillen. (Top_{cg})_{Quillen} \underoverset \underset{(-)^X}{\longrightarrow} \overset{X \times (-)}{\longleftarrow} {\bot} (Top_{cg})_{Quillen} \,.
Proof

By example we have that the left adjoint functor is equivalently the pushout product functor with the initial morphism of XX:

X×()(X)(). X \times (-) \simeq (\emptyset \to X) \Box (-) \,.

By assumption (X)(\emptyset \to X) is a cofibration, and hence prop. says that this is a left Quillen functor.

The statement and proof of prop. has a direct analogue in pointed topological spaces

Proposition

For X(Top cg */) QuillenX \in (Top^{\ast/}_{cg})_{Quillen} cofibrant with respect to the classical model structure on pointed compactly generated topological spaces (theorem , prop. ) (hence a retract of a cell complex with non-degenerate basepoint, remark ) then the pointed product-hom-adjunction from corollary is a Quillen adjunction (def. ):

(Top cg */) QuillenMaps(X,) *X()(Top cg */) Quillen. (Top^{\ast/}_{cg})_{Quillen} \underoverset \underset{Maps(X,-)_\ast}{\longrightarrow} \overset{X \wedge (-)}{\longleftarrow} {\bot} (Top^{\ast/}_{cg})_{Quillen} \,.
Proof

Let now \Box_\wedge denote the smash pushout product and () ()(-)^{\Box(-)} the smash pullback powering defined as in def. and def. , but with Cartesian product replaced by smash product (def. ) and compactly generated mapping space replaced by pointed mapping spaces (def. ).

By theorem (Top cg */) Quillen(Top_{cg}^{\ast/})_{Quillen} is cofibrantly generated by I Top */=(I Top) +I_{Top^{\ast/}} = (I_{Top})_+ and J Top */=(J Top) +J_{Top^{\ast/}}= (J_{Top})_+. Example gives that for i nI Topi_n \in I_{Top} and j nJ Topj_n \in J_{Top} then

(i n 1) + (i n 2) +(i n 1+n 2) + (i_{n_1})_+ \Box_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+

and

(i n 1) + (i n 2) +(i n 1+n 2) +. (i_{n_1})_+ \wedge_\wedge (i_{n_2})_+ \simeq (i_{n_1 + n_2})_+ \,.

Hence the pointed analog of prop. holds and therefore so does the pointed analog of the conclusion in prop. .

Model structure on topological functors

With classical topological homotopy theory in hand (theorem , theorem ), it is straightforward now to generalize this to a homotopy theory of topological diagrams. This is going to be the basis for the stable homotopy theory of spectra, because spectra may be identified with certain topological diagrams (prop.).

Technically, “topological diagram” here means “Top-enriched functor”. We now discuss what this means and then observe that as an immediate corollary of theorem we obtain a model category structure on topological diagrams.

As a by-product, we obtain the model category theory of homotopy colimits in topological spaces, which will be useful.

In the following we say Top-enriched category and Top-enriched functor etc. for what often is referred to as “topological category” and “topological functor” etc. As discussed there, these latter terms are ambiguous.

Literature (Riehl, chapter 3) for basics of enriched category theory; (Piacenza 91) for the projective model structure on topological functors.

\,

Definition

A topologically enriched category 𝒞\mathcal{C} is a Top cgTop_{cg}-enriched category, hence:

  1. a class Obj(𝒞)Obj(\mathcal{C}), called the class of objects;

  2. for each a,bObj(𝒞)a, b\in Obj(\mathcal{C}) a compactly generated topological space (def. )

    𝒞(a,b)Top cg, \mathcal{C}(a,b)\in Top_{cg} \,,

    called the space of morphisms or the hom-space between aa and bb;

  3. for each a,b,cObj(𝒞)a,b,c\in Obj(\mathcal{C}) a continuous function

    a,b,c:𝒞(a,b)×𝒞(b,c)𝒞(a,c) \circ_{a,b,c} \;\colon\; \mathcal{C}(a,b)\times \mathcal{C}(b,c) \longrightarrow \mathcal{C}(a,c)

    out of the cartesian product (by cor. : the image under kk of the product topological space), called the composition operation;

  4. for each aObj(𝒞)a \in Obj(\mathcal{C}) a point Id a𝒞(a,a)Id_a\in \mathcal{C}(a,a), called the identity morphism on aa

such that the composition is associative and unital.

Similarly a pointed topologically enriched category is such a structure with Top cgTop_{cg} replaced by Top cg */Top^{\ast/}_{cg} (def. ) and with the Cartesian product replaced by the smash product (def. ) of pointed topological spaces.

Remark

Given a (pointed) topologically enriched category as in def. , then forgetting the topology on the hom-spaces (along the forgetful functor U:Top cgSetU \colon Top_{cg} \to Set) yields an ordinary locally small category with

Hom 𝒞(a,b)=U(𝒞(a,b)). Hom_{\mathcal{C}}(a,b) = U(\mathcal{C}(a,b)) \,.

It is in this sense that 𝒞\mathcal{C} is a category with extra structure, and hence “enriched”.

The archetypical example is Top cgTop_{cg} itself:

Example

The category Top cgTop_{cg} (def. ) canonically obtains the structure of a topologically enriched category, def. , with hom-spaces given by the compactly generated mapping spaces (def. )

Top cg(X,Y)Y X Top_{cg}(X,Y) \coloneqq Y^X

and with composition

Y X×Z YZ X Y^X \times Z^Y \longrightarrow Z^X

given by the adjunct under the (product\dashv mapping-space)-adjunction from prop. of the evaluation morphisms

X×Y X×Z Y(ev,id)Y×Z YevZ. X \times Y^X \times Z^Y \overset{(ev, id)}{\longrightarrow} Y \times Z^Y \overset{ev}{\longrightarrow} Z \,.

Similarly, pointed compactly generated topological spaces Top k */Top_k^{\ast/} form a pointed topologically enriched category, using the pointed mapping spaces from example :

Top cg */(X,Y)Maps(X,Y) *. Top^{\ast/}_{cg}(X,Y) \coloneqq Maps(X,Y)_\ast \,.
Definition

A topologically enriched functor between two topologically enriched categories

F:𝒞𝒟 F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}

is a Top cgTop_{cg}-enriched functor, hence:

  1. a function

    F 0:Obj(𝒞)Obj(𝒟) F_0 \;\colon\; Obj(\mathcal{C}) \longrightarrow Obj(\mathcal{D})

    of objects;

  2. for each a,bObj(𝒞)a,b \in Obj(\mathcal{C}) a continuous function

    F a,b:𝒞(a,b)𝒟(F 0(a),F 0(b)) F_{a,b} \;\colon\; \mathcal{C}(a,b) \longrightarrow \mathcal{D}(F_0(a), F_0(b))

    of hom-spaces,

such that this preserves composition and identity morphisms in the evident sense.

A homomorphism of topologically enriched functors

η:FG \eta \;\colon\; F \Rightarrow G

is a Top cgTop_{cg}-enriched natural transformation: for each cObj(𝒞)c \in Obj(\mathcal{C}) a choice of morphism η c𝒟(F(c),G(c))\eta_c \in \mathcal{D}(F(c),G(c)) such that for each pair of objects c,d𝒞c,d \in \mathcal{C} the two continuous functions

η dF():𝒞(c,d)𝒟(F(c),G(d)) \eta_d \circ F(-) \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))

and

G()η c:𝒞(c,d)𝒟(F(c),G(d)) G(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow \mathcal{D}(F(c), G(d))

agree.

We write [𝒞,𝒟][\mathcal{C}, \mathcal{D}] for the resulting category of topologically enriched functors.

Remark

The condition on an enriched natural transformation in def. is just that on an ordinary natural transformation on the underlying unenriched functors, saying that for every morphisms f:cdf \colon c \to d there is a commuting square

f𝒞(c,c)×X η c F(c) 𝒞(c,f) F(f) 𝒞(c,d)×X η d F(d). f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) } \,.
Example

For 𝒞\mathcal{C} any topologically enriched category, def. then a topologically enriched functor (def. )

F:𝒞Top cg F \;\colon\; \mathcal{C} \longrightarrow Top_{cg}

to the archetypical topologically enriched category from example may be thought of as a topologically enriched copresheaf, at least if 𝒞\mathcal{C} is small (in that its class of objects is a proper set).

Hence the category of topologically enriched functors

[𝒞,Top cg] [\mathcal{C}, Top_{cg}]

according to def. may be thought of as the (co-)presheaf category over 𝒞\mathcal{C} in the realm of topological enriched categories.

A functor F[𝒞,Top cg]F \in [\mathcal{C}, Top_{cg}] is equivalently

  1. a compactly generated topological space F aTop cgF_a \in Top_{cg} for each object aObj(𝒞)a \in Obj(\mathcal{C});

  2. a continuous function

    F a×𝒞(a,b)F b F_a \times \mathcal{C}(a,b) \longrightarrow F_b

    for all pairs of objects a,bObj(𝒞)a,b \in Obj(\mathcal{C})

such that composition is respected, in the evident sense.

For every object c𝒞c \in \mathcal{C}, there is a topologically enriched representable functor, denoted y(c)y(c) or 𝒞(c,)\mathcal{C}(c,-) which sends objects to

y(c)(d)=𝒞(c,d)Top cg y(c)(d) = \mathcal{C}(c,d) \in Top_{cg}

and whose action on morphisms is, under the above identification, just the composition operation in 𝒞\mathcal{C}.

Proposition

For 𝒞\mathcal{C} any small topologically enriched category, def. then the enriched functor category [𝒞,Top cg][\mathcal{C}, Top_{cg}] from example has all limits and colimits, and they are computed objectwise:

if

F :I[𝒞,Top cg] F_\bullet \;\colon\; I \longrightarrow [\mathcal{C}, Top_{cg}]

is a diagram of functors and c𝒞c\in \mathcal{C} is any object, then

(lim iF i)(c)lim i(F i(c))Top cg (\underset{\longleftarrow}{\lim}_i F_i)(c) \simeq \underset{\longleftarrow}{\lim}_i (F_i(c)) \;\;\in Top_{cg}

and

(lim iF i)(c)lim i(F i(c))Top cg. (\underset{\longrightarrow}{\lim}_i F_i)(c) \simeq \underset{\longrightarrow}{\lim}_i (F_i(c)) \;\; \in Top_{cg} \,.
Proof

First consider the underlying diagram of functors F i F_i^\circ where the topology on the hom-spaces of 𝒞\mathcal{C} and of Top cgTop_{cg} has been forgotten. Then one finds

(lim iF i )(c)lim i(F i (c))Set (\underset{\longleftarrow}{\lim}_i F^\circ_i)(c) \simeq \underset{\longleftarrow}{\lim}_i (F^\circ_i(c)) \;\;\in Set

and

(lim iF i )(c)lim i(F i (c))Set (\underset{\longrightarrow}{\lim}_i F^\circ _i)(c) \simeq \underset{\longrightarrow}{\lim}_i (F^\circ_i(c)) \;\; \in Set

by the universal property of limits and colimits. (Given a morphism of diagrams then a unique compatible morphism between their limits or colimits, respectively, is induced as the universal factorization of the morphism of diagrams regarded as a cone or cocone, respectvely, over the codomain or domain diagram, respectively).

Hence it only remains to see that equipped with topology, these limits and colimits in SetSet become limits and colimits in Top cgTop_{cg}. That is just the statement of prop. with corollary .

Definition

Let 𝒞\mathcal{C} be a topologically enriched category, def. , with [𝒞,Top cg][\mathcal{C}, Top_{cg}] its category of topologically enriched copresheaves from example .

  1. Define a functor

    ()():[𝒞,Top cg]×Top cg[𝒞,Top cg] (-)\cdot(-) \;\colon\; [\mathcal{C}, Top_{cg}] \times Top_{cg} \longrightarrow [\mathcal{C}, Top_{cg}]

    by forming objectwise cartesian products (hence kk of product topological spaces)

    FX:cF(c)×X. F \cdot X \;\colon\; c \mapsto F(c) \times X \,.

    This is called the tensoring of [𝒞,Top cg][\mathcal{C},Top_{cg}] over Top cgTop_{cg} (Def. ).

  2. Define a functor

    () ():(Top cg) op×[𝒞,Top cg][𝒞,Top cg] (-)^{(-)} \;\colon\; (Top_{cg})^{op} \times [\mathcal{C}, Top_{cg}] \longrightarrow [\mathcal{C}, Top_{cg}]

    by forming objectwise compactly generated mapping spaces (def. )

    F X:cF(c) X. F^X \;\colon\; c \mapsto F(c)^X \,.

    This is called the powering of [𝒞,Top cg][\mathcal{C}, Top_{cg}] over Top cgTop_{cg}.

Analogously, for 𝒞\mathcal{C} a pointed topologically enriched category, def. , with [𝒞,Top cg */][\mathcal{C}, Top_{cg}^{\ast/}] its category of pointed topologically enriched copresheaves from example , then:

  1. Define a functor

    ()():[𝒞,Top cg */]×Top cg */[𝒞,Top cg */] (-)\wedge(-) \;\colon\; [\mathcal{C}, Top^{\ast/}_{cg}] \times Top^{\ast/}_{cg} \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]

    by forming objectwise smash products (def. )

    FX:cF(c)X. F \wedge X \;\colon\; c \mapsto F(c) \wedge X \,.

    This is called the smash tensoring of [𝒞,Top cg */][\mathcal{C},Top^{\ast/}_{cg}] over Top cg */Top^{\ast/}_{cg} (Def. ).

  2. Define a functor

    Maps(,) *:Top cg */×[𝒞,Top cg */][𝒞,Top cg */] Maps(-,-)_\ast \;\colon\; Top^{\ast/}_{cg} \times [\mathcal{C}, Top^{\ast/}_{cg}] \longrightarrow [\mathcal{C}, Top^{\ast/}_{cg}]

    by forming objectwise pointed mapping spaces (example )

    F X:cMaps(X,F(c)) *. F^X \;\colon\; c \mapsto Maps(X,F(c))_\ast \,.

    This is called the pointed powering of [𝒞,Top cg][\mathcal{C}, Top_{cg}] over Top cgTop_{cg}.

There is a full blown Top cgTop_{cg}-enriched Yoneda lemma. The following records a slightly simplified version which is all that is needed here:

Proposition

(topologically enriched Yoneda-lemma)

Let 𝒞\mathcal{C} be a topologically enriched category, def. , write [𝒞,Top cg][\mathcal{C}, Top_{cg}] for its category of topologically enriched (co-)presheaves, and for cObj(𝒞)c\in Obj(\mathcal{C}) write y(c)=𝒞(c,)[𝒞,Top k]y(c) = \mathcal{C}(c,-) \in [\mathcal{C}, Top_k] for the topologically enriched functor that it represents, all according to example . Recall the tensoring operation (F,X)FX(F,X) \mapsto F \cdot X from def. .

For cObj(𝒞)c\in Obj(\mathcal{C}), XTop cgX \in Top_{cg} and F[𝒞,Top cg]F \in [\mathcal{C}, Top_{cg}], there is a natural bijection between

  1. morphisms y(c)XFy(c) \cdot X \longrightarrow F in [𝒞,Top cg][\mathcal{C}, Top_{cg}];

  2. morphisms XF(c)X \longrightarrow F(c) in Top cgTop_{cg}.

In short:

y(c)XFXF(c) \frac{ y(c)\cdot X \longrightarrow F }{ X \longrightarrow F(c) }
Proof

Given a morphism η:y(c)XF\eta \colon y(c) \cdot X \longrightarrow F consider its component

η c:𝒞(c,c)×XF(c) \eta_c \;\colon\; \mathcal{C}(c,c)\times X \longrightarrow F(c)

and restrict that to the identity morphism id c𝒞(c,c)id_c \in \mathcal{C}(c,c) in the first argument

η c(id c,):XF(c). \eta_c(id_c,-) \;\colon\; X \longrightarrow F(c) \,.

We claim that just this η c(id c,)\eta_c(id_c,-) already uniquely determines all components

η d:𝒞(c,d)×XF(d) \eta_d \;\colon\; \mathcal{C}(c,d)\times X \longrightarrow F(d)

of η\eta, for all dObj(𝒞)d \in Obj(\mathcal{C}): By definition of the transformation η\eta (def. ), the two functions

F()η c:𝒞(c,d)F(d) 𝒞(c,c)×X F(-) \circ \eta_c \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}

and

η d𝒞(c,)×X:𝒞(c,d)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,-) \times X \;\colon\; \mathcal{C}(c,d) \longrightarrow F(d)^{\mathcal{C}(c,c) \times X}

agree. This means (remark ) that they may be thought of jointly as a function with values in commuting squares in Top cgTop_{cg} of this form:

f𝒞(c,c)×X η c F(c) 𝒞(c,f) F(f) 𝒞(c,d)×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \mathcal{C}(c,c) \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \mathcal{C}(c,d) \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

For any f𝒞(c,d)f \in \mathcal{C}(c,d), consider the restriction of

η d𝒞(c,f)F(d) 𝒞(c,c)×X \eta_d \circ \mathcal{C}(c,f) \in F(d)^{\mathcal{C}(c,c) \times X}

to id c𝒞(c,c)id_c \in \mathcal{C}(c,c), hence restricting the above commuting squares to

f{id c}×X η c F(c) 𝒞(c,f) F(f) {f}×X η d F(d) f \;\;\;\; \mapsto \;\;\;\; \array{ \{id_c\} \times X &\overset{\eta_c}{\longrightarrow}& F(c) \\ {}^{\mathllap{\mathcal{C}(c,f)}}\downarrow && \downarrow^{\mathrlap{F(f)}} \\ \{f\} \times X &\underset{\eta_d}{\longrightarrow}& F(d) }

This shows that η d\eta_d is fixed to be the function

η d(f,x)=F(f)η c(id c,x) \eta_d(f,x) = F(f)\circ \eta_c(id_c,x)

and this is a continuous function since all the operations it is built from are continuous.

Conversely, given a continuous function α:XF(c)\alpha \colon X \longrightarrow F(c), define for each dd the function

η d:(f,x)F(f)α. \eta_d \colon (f,x) \mapsto F(f) \circ \alpha \,.

Running the above analysis backwards shows that this determines a transformation η:y(c)×XF\eta \colon y(c)\times X \to F.

Definition

For 𝒞\mathcal{C} a small topologically enriched category, def. , write

I Top 𝒞{y(c)(S n1ι nD n)} n,cObj(𝒞) I_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (S^{n-1} \overset{\iota_n}{\longrightarrow} D^n) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}}

and

J Top 𝒞{y(c)(D n(id,δ 0)D n×I)} n,cObj(𝒞) J_{Top}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c)\cdot (D^n \overset{(id, \delta_0)}{\longrightarrow} D^n \times I) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}}

for the sets of morphisms given by tensoring (def. ) the representable functors (example ) with the generating cofibrations (def.) and acyclic generating cofibrations (def. ), respectively, of (Top cg) Quillen(Top_{cg})_{Quillen} (theorem ).

These are going to be called the generating cofibrations and acyclic generating cofibrations for the projective model structure on topologically enriched functors over 𝒞\mathcal{C}.

Analgously, for 𝒞\mathcal{C} a pointed topologically enriched category, write

I Top */ 𝒞{y(c)(S + n1(ι n) +D + n)} n,cObj(𝒞) I_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c) \wedge (S^{n-1}_+ \overset{(\iota_n)_+}{\longrightarrow} D^n_+) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}}

and

J Top */ 𝒞{y(c)(D + n(id,δ 0) +(D n×I) +)} n,cObj(𝒞) J_{Top^{\ast/}}^{\mathcal{C}} \;\coloneqq\; \left\{ y(c) \wedge (D^n_+ \overset{(id, \delta_0)_+}{\longrightarrow} (D^n \times I)_+) \right\}_{{n \in \mathbb{N},} \atop {c \in Obj(\mathcal{C})}}

for the analogous construction applied to the pointed generating (acyclic) cofibrations of def. .

Definition

Given a small (pointed) topologically enriched category 𝒞\mathcal{C}, def. , say that a morphism in the category of (pointed) topologically enriched copresheaves [𝒞,Top cg][\mathcal{C}, Top_{cg}] ([𝒞,Top cg */][\mathcal{C},Top_{cg}^{\ast/}]), example , hence a natural transformation between topologically enriched functors, η:FG\eta \colon F \to G is

  • a projective weak equivalence, if for all cObj(𝒞)c\in Obj(\mathcal{C}) the component η c:F(c)G(c)\eta_c \colon F(c) \to G(c) is a weak homotopy equivalence (def. );

  • a projective fibration if for all cObj(𝒞)c\in Obj(\mathcal{C}) the component η c:F(c)G(c)\eta_c \colon F(c) \to G(c) is a Serre fibration (def. );

  • a projective cofibration if it is a retract (rmk. ) of an I Top 𝒞I_{Top}^{\mathcal{C}}-relative cell complex (def. , def. ).

Write

[𝒞,(Top cg) Quillen] proj [\mathcal{C}, (Top_{cg})_{Quillen}]_{proj}

and

[𝒞,(Top cg */) Quillen] proj [\mathcal{C}, (Top^{\ast/}_{cg})_{Quillen}]_{proj}

for the categories of topologically enriched functors equipped with these classes of morphisms.

Theorem

The classes of morphisms in def. constitute a model category structure on [𝒞,Top cg][\mathcal{C}, Top_{cg}] and [𝒞,Top cg */][\mathcal{C}, Top^{\ast/}_{cg}], called the projective model structure on enriched functors

[𝒞,(Top cg) Quillen] proj [\mathcal{C}, (Top_{cg})_{Quillen}]_{proj}

and

[𝒞,(Top cg */) Quillen] proj [\mathcal{C}, (Top^{\ast/}_{cg})_{Quillen}]_{proj}

These are cofibrantly generated model category, def. , with set of generating (acyclic) cofibrations the sets I Top 𝒞I_{Top}^{\mathcal{C}}, J Top 𝒞J_{Top}^{\mathcal{C}} and I Top */ 𝒞I_{Top^{\ast/}}^{\mathcal{C}}, J Top */ 𝒞J_{Top^{\ast/}}^{\mathcal{C}} from def. , respectively.

(Piacenza 91, theorem 5.4)

Proof

By prop. the category has all limits and colimits, hence it remains to check the model structure

But via the enriched Yoneda lemma (prop. ) it follows that proving the model structure reduces objectwise to the proof of theorem , theorem . In particular, the technical lemmas , and generalize immediately to the present situation, with the evident small change of wording:

For instance, the fact that a morphism of topologically enriched functors η:FG\eta \colon F \to G that has the right lifting property against the elements of I Top 𝒞I_{Top}^{\mathcal{C}} is a projective weak equivalence, follows by noticing that for fixed η:FG\eta \colon F \to G the enriched Yoneda lemma prop. gives a natural bijection of commuting diagrams (and their fillers) of the form

(y(c)S n1 F (idι n) η y(c)D n G)(S n1 F(c) η c D n G(c)), \left( \array{ y(c) \cdot S^{n-1} &\longrightarrow& F \\ {}^{\mathllap{(id\cdot \iota_n)}}\downarrow && \downarrow^{\mathrlap{\eta}} \\ y(c) \cdot D^n &\longrightarrow& G } \right) \;\;\;\leftrightarrow\;\;\; \left( \array{ S^{n-1} &\longrightarrow& F(c) \\ \downarrow && \downarrow^{\mathrlap{\eta_c}} \\ D^n &\longrightarrow& G(c) } \right) \,,

and hence the statement follows with part A) of the proof of lemma .

With these three lemmas in hand, the remaining formal part of the proof goes through verbatim as above: repeatedly use the small object argument (prop. ) and the retract argument (prop. ) to establish the two weak factorization systems. (While again the structure of a category with weak equivalences is evident.)

Example

Given examples and , the next evident example of a pointed topologically enriched category besides Top cg */Top^{\ast/}_{cg} itself is the functor category

[Top cg */,Top cg */]. [Top_{cg}^{\ast/}, Top_{cg}^{\ast/}] \,.

The only technical problem with this is that Top cg */Top^{\ast/}_{cg} is not a small category (it has a proper class of objects), which means that the existence of all limits and colimits via prop. may (and does) fail.

But so we just restrict to a small topologically enriched subcategory. A good choice is the full subcategory

Top cg,fin */Top cg */ Top^{\ast/}_{cg,fin} \hookrightarrow Top^{\ast/}_{cg}

of topological spaces homoemorphic to finite CW-complexes. The resulting projective model category (via theorem )

[Top cg,fin */,(Top cg */) Quillen] proj [Top_{cg,fin}^{\ast/}\;,\; (Top^{\ast/}_{cg})_{Quillen}]_{proj}

is also also known as the strict model structure for excisive functors. (This terminology is the special case for n=1n = 1 of the terminology “n-excisive functors” as used in “Goodwillie calculus”, a homotopy-theoretic analog of differential calculus.) After enlarging its class of weak equivalences while keeping the cofibrations fixed, this will become Quillen equivalent to a model structure for spectra. This we discuss in part 1.2, in the section on pre-excisive functors.

One consequence of theorem is the model category theoretic incarnation of the theory of homotopy colimits.

Observe that ordinary limits and colimits (def. ) are equivalently characterized in terms of adjoint functors:

Let 𝒞\mathcal{C} be any category and let II be a small category. Write [I,𝒞][I,\mathcal{C}] for the corresponding functor category. We may think of its objects as II-shaped diagrams in 𝒞\mathcal{C}, and of its morphisms as homomorphisms of these diagrams. There is a canonical functor

const I:𝒞[I,𝒞] const_I \;\colon\; \mathcal{C} \overset{}{\longrightarrow} [I,\mathcal{C}]

which sends each object of 𝒞\mathcal{C} to the diagram that is constant on this object. Inspection of the definition of the universal properties of limits and colimits on one hand, and of left adjoint and right adjoint functors on the other hand, shows that

  1. precisely when 𝒞\mathcal{C} has all colimits of shape II, then the functor const Iconst_I has a left adjoint functor, which is the operation of forming these colimits:

    [I,𝒞]const Ilim I𝒞 [I,\mathcal{C}] \underoverset {\underset{const_I}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}_I}{\longrightarrow}} {\bot} \mathcal{C}
  2. precisely when 𝒞\mathcal{C} has all limits of shape II, then the functor const Iconst_I has a right adjoint functor, which is the operation of forming these limits.

    [I,𝒞]lim Iconst I𝒞 [I,\mathcal{C}] \underoverset {\underset{\underset{\longleftarrow}{\lim}_I}{\longrightarrow}} {\overset{const_I}{\longleftarrow}} {\bot} \mathcal{C}
Proposition

Let II be a small topologically enriched category (def. ). Then the (lim Iconst I)(\underset{\longrightarrow}{\lim}_I \dashv const_I)-adjunction

[I,(Top cg) Quillen] projconst Ilim I(Top cg) Quillen [I,(Top_{cg})_{Quillen}]_{proj} \underoverset {\underset{const_I}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}_I}{\longrightarrow}} {\bot} (Top_{cg})_{Quillen}

is a Quillen adjunction (def. ) between the projective model structure on topological functors on II, from theorem , and the classical model structure on topological spaces from theorem .

Similarly, if II is enriched in pointed topological spaces, then for the classical model structure on pointed topological spaces (prop. , theorem ) the adjunction

[I,(Top cg */) Quillen] projconstlim(Top cg */) Quillen [I,(Top^{\ast/}_{cg})_{Quillen}]_{proj} \underoverset {\underset{const}{\longleftarrow}} {\overset{\underset{\longrightarrow}{\lim}}{\longrightarrow}} {\bot} (Top^{\ast/}_{cg})_{Quillen}

is a Quillen adjunction.

Proof

Since the fibrations and weak equivalences in the projective model structure (def. ) on the functor category are objectwise those of (Top cg) Quillen(Top_{cg})_{Quillen} and of (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}, respectively, it is immediate that the functor const Iconst_I preserves these. In particular it preserves fibrations and acyclic fibrations and so the claim follows (prop. ).

Definition

(homotopy colimit)

In the situation of prop. we say that the left derived functor (def. ) of the colimit functor is the homotopy colimit

hocolim I𝕃lim I:Ho([I,Top])Ho(Top) hocolim_I \coloneqq \mathbb{L}\underset{\longrightarrow}{\lim}_I \;\colon\; Ho([I,Top]) \longrightarrow Ho(Top)

and

hocolim I𝕃lim I:Ho([I,Top */])Ho(Top */). hocolim_I \coloneqq \mathbb{L}\underset{\longrightarrow}{\lim}_I \;\colon\; Ho([I,Top^{\ast/}]) \longrightarrow Ho(Top^{\ast/}) \,.
Remark

Since every object in (Top cg) Quillen(Top_{cg})_{Quillen} and in (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} is fibrant, the homotopy colimit of any diagram X X_\bullet, according to def. , is (up to weak homotopy equivalence) the result of forming the ordinary colimit of any projectively cofibrant replacement X^ W projX \hat X_\bullet \overset{\in W_{proj}}{\to} X_\bullet.

Example

Write \mathbb{N}^{\leq} for the poset (def. ) of natural numbers, hence for the small category (with at most one morphism from any given object to any other given object) that looks like

={0123}. \mathbb{N}^{\leq} = \left\{ 0 \to 1 \to 2 \to 3 \to \cdots \right\} \,.

Regard this as a topologically enriched category with the, necessarily, discrete topology on its hom-sets.

Then a topologically enriched functor

X : Top cg X_\bullet \;\colon\; \mathbb{N}^{\leq} \longrightarrow Top_{cg}

is just a plain functor and is equivalently a sequence of continuous functions (morphisms in Top cgTop_{cg}) of the form (also called a cotower)

X 0f 0X 1f 1X 2f 2X 3. X_0 \overset{f_0}{\longrightarrow} X_1 \overset{f_1}{\longrightarrow} X_2 \overset{f_2}{\longrightarrow} X_3 \longrightarrow \cdots \,.

It is immediate to check that those sequences X X_\bullet which are cofibrant in the projective model structure (theorem ) are precisely those for which

  1. all component morphisms f if_i are cofibrations in (Top cg) Quillen(Top_{cg})_{Quillen} or (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen}, respectively, hence retracts (remark ) of relative cell complex inclusions (def. );

  2. the object X 0X_0, and hence all other objects, are cofibrant, hence are retracts of cell complexes (def. ).

By example it is immediate that the operation of forming colimits sends projective (acyclic) cofibrations between sequences of topological spaces to (acyclic) cofibrations in the classical model structure on pointed topological spaces. On those projectively cofibrant sequences where every map is not just a retract of a relative cell complex inclusion, but a plain relative cell complex inclusion, more is true:

Proposition

In the projective model structures on cotowers in topological spaces, [ ,(Top cg) Quillen] proj[\mathbb{N}^{\leq}, (Top_{cg})_{Quillen}]_{proj} and [ ,(Top cg */) Quillen] proj[\mathbb{N}^{\leq}, (Top^{\ast/}_{cg})_{Quillen}]_{proj} from def. , the following holds:

  1. The colimit functor preserves fibrations between sequences of relative cell complex inclusions;

  2. Let II be a finite category, let D ():I[ ,Top cg]D_\bullet(-) \colon I \to [\mathbb{N}^{\leq}, Top_{cg}] be a finite diagram of sequences of relative cell complexes. Then there is a weak homotopy equivalence

    lim n(lim iD n(i))W cllim i(lim nD n(i)) \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i D_n(i) \right) \overset{\in W_{cl}}{\longrightarrow} \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} D_n(i) \right)

    from the colimit over the limit sequnce to the limit of the colimits of sequences.

Proof

Regarding the first statement:

Use that both (Top cg) Quillen(Top_{cg})_{Quillen} and (Top cg */) Quillen(Top^{\ast/}_{cg})_{Quillen} are cofibrantly generated model categories (theorem ) whose generating acyclic cofibrations have compact topological spaces as domains and codomains. The colimit over a sequence of relative cell complexes (being a transfinite composition) yields another relative cell complex, and hence lemma says that every morphism out of the domain or codomain of a generating acyclic cofibration into this colimit factors through a finite stage inclusion. Since a projective fibration is a degreewise fibration, we have the lifting property at that finite stage, and hence also the lifting property against the morphisms of colimits.

Regarding the second statement:

This is a model category theoretic version of a standard fact of plain category theory, which says that in the category Set of sets, filtered colimits commute with finite limits in that there is an isomorphism of sets of the form which we have to prove is a weak homotopy equivalence of topological spaces. But now using that weak homotopy equivalences are detected by forming homotopy groups (def. ), hence hom-sets out of n-spheres, and since nn-spheres are compact topological spaces, lemma says that homming out of nn-spheres commutes over the colimits in question. Moreover, generally homming out of anything commutes over limits, in particular finite limits (every hom functor is left exact functor in the second variable). Therefore we find isomorphisms of the form

Hom(S q,lim n(lim iD n(i)))lim n(lim iHom(S q,D n(i)))lim i(lim nHom(S qD n(i)))Hom(S q,lim i(lim nD n(i))) Hom\left( S^q, \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i D_n(i) \right) \right) \simeq \underset{\longrightarrow}{\lim}_{n} \left( \underset{\longleftarrow}{\lim}_i Hom\left(S^q, D_n(i)\right) \right) \overset{\sim}{\longrightarrow} \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} Hom\left(S^q D_n(i)\right) \right) \simeq Hom\left( S^q, \underset{\longleftarrow}{\lim}_i \left( \underset{\longrightarrow}{\lim}_{n} D_n(i) \right) \right)

and similarly for the left homotopies Hom(S q×I,)Hom(S^q \times I,-) (and similarly for the pointed case). This implies the claimed isomorphism on homotopy groups.

\,

\infty-Groupoids II): Simplicial homotopy theory

With groupoids and chain complexes we have seen two kinds of objects which support concepts of homotopy theory, such as a concept of homotopy equivalence between them (equivalence of groupoids on the one hand, and quasi-isomorphism on the other). In some sense these two cases are opposite extremes in the more general context of homotopy theory:

  • chain complexes have homotopical structure (e.g. chain homology) in arbitrary high degree, i.e. they may be homotopy n-types for arbitrary nn, but they are fully abelian in that there is never any nonabelian group structure in a chain complex, not is there any non-trivial action of the homology groups of a chain complex on each other;

  • groupoids have more general non-abelian structure, for every (nonabelian) group there is a groupoid which has this as its fundamental group, but this fundamental group (in degree 1) is already the highest homotopical structure they carry, groupoids are necessarily homotopy 1-types.

On the other hand, both groupoids and chain complexes naturally have incarnations in the joint context of simplicial sets. We now discuss how their common joint generalization is given by those simplicial sets whose simplices have a sensible notion of composition and inverses, the Kan complexes.

topologicalspaces higherpathgroupoid groupoids Grothendiecknerve Kancomplexesgroupoids DoldKancorrespondence chaincomplexes includedin simplicialsets \array{ && {topological \atop spaces} \\ && \downarrow^{\mathrlap{{higher \atop path}\atop groupoid}} & \\ groupoids &\stackrel{{Grothendieck \atop nerve}}{\longrightarrow}& { {\mathbf{Kan}\;\mathbf{complexes}} \atop {\simeq \infty-groupoids} } &\stackrel{{Dold-Kan \atop correspondence}}{\longleftarrow}& {chain \atop complexes} \\ && \downarrow^{\mathrlap{included \atop in}} \\ && {simplicial \atop sets} }

Kan complexes serve as a standard powerful model on which the complete formulation of homotopy theory (without geometry) may be formulated.

Simplicial sets

The concept of simplicial sets is secretly well familiar already in basic algebraic topology: it reflects just the abstract structure carried by the singular simplicial complexes of topological spaces, as in the definition of singular homology and singular cohomology.

Conversely, every simplicial set may be geometrically realized as a topological space. These two adjoint operations turn out to exhibit the homotopy theory of simplicial sets as being equivalent (Quillen equivalent) to the homotopy theory of topological spaces. For some purposes, working in simplicial homotopy theory is preferable over working with topological homotopy theory.

Definition

(topological simplex)

For nn \in \mathbb{N}, the topological n-simplex is, up to homeomorphism, the topological space whose underlying set is the subset

Δ n{x n+1| i=0 nx i=1andi.x i0} n+1 \Delta^n \coloneqq \{ \vec x \in \mathbb{R}^{n+1} | \sum_{i = 0 }^n x_i = 1 \; and \; \forall i . x_i \geq 0 \} \subset \mathbb{R}^{n+1}

of the Cartesian space n+1\mathbb{R}^{n+1}, and whose topology is the subspace topology induces from the canonical topology in n+1\mathbb{R}^{n+1}.

Example

For n=0n = 0 this is the point, Δ 0=*\Delta^0 = *.

For n=1n = 1 this is the standard interval object Δ 1=[0,1]\Delta^1 = [0,1].

For n=2n = 2 this is the filled triangle.

For n=3n = 3 this is the filled tetrahedron.

Definition

For nn \in \mathbb{N}, n1\n \geq 1 and 0kn0 \leq k \leq n, the kkth (n1)(n-1)-face (inclusion) of the topological nn-simplex, def. , is the subspace inclusion

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \hookrightarrow \Delta^n

induced under the coordinate presentation of def. , by the inclusion

n n+1 \mathbb{R}^n \hookrightarrow \mathbb{R}^{n+1}

which “omits” the kkth canonical coordinate:

(x 0,,x n1)(x 0,,x k1,0,x k,,x n). (x_0, \cdots , x_{n-1}) \mapsto (x_0, \cdots, x_{k-1} , 0 , x_{k}, \cdots, x_n) \,.
Example

The inclusion

δ 0:Δ 0Δ 1 \delta_0 : \Delta^0 \to \Delta^1

is the inclusion

{1}[0,1] \{1\} \hookrightarrow [0,1]

of the “right” end of the standard interval. The other inclusion

δ 1:Δ 0Δ 1 \delta_1 : \Delta^0 \to \Delta^1

is that of the “left” end {0}[0,1]\{0\} \hookrightarrow [0,1].

(graphics taken from Friedman 08)

Definition

For nn \in \mathbb{N} and 0k<n0 \leq k \lt n the kkth degenerate (n)(n)-simplex (projection) is the surjective map

σ k:Δ nΔ n1 \sigma_k : \Delta^{n} \to \Delta^{n-1}

induced under the barycentric coordinates of def. under the surjection

n+1 n \mathbb{R}^{n+1} \to \mathbb{R}^n

which sends

(x 0,,x n)(x 0,,x k+x k+1,,x n). (x_0, \cdots, x_n) \mapsto (x_0, \cdots, x_{k} + x_{k+1}, \cdots, x_n) \,.
Definition

(singular simplex)

For XX \in Top and nn \in \mathbb{N}, a singular nn-simplex in XX is a continuous map

σ:Δ nX \sigma : \Delta^n \to X

from the topological nn-simplex, def. , to XX.

Write

(SingX) nHom Top(Δ n,X) (Sing X)_n \coloneqq Hom_{Top}(\Delta^n , X)

for the set of singular nn-simplices of XX.

(graphics taken from Friedman 08)

The sets (SingX) (Sing X)_\bullet here are closely related by an interlocking system of maps that make them form what is called a simplicial set, and as such the collection of these sets of singular simplices is called the singular simplicial complex of XX. We discuss the definition of simplicial sets now and then come back to this below in def. .

Since the topological nn-simplices Δ n\Delta^n from def. sit inside each other by the face inclusions of def.

δ k:Δ n1Δ n \delta_k : \Delta^{n-1} \to \Delta^{n}

and project onto each other by the degeneracy maps, def.

σ k:Δ n+1Δ n \sigma_k : \Delta^{n+1} \to \Delta^n

we dually have functions

d kHom Top(δ k,X):(SingX) n(SingX) n1 d_k \coloneqq Hom_{Top}(\delta_k, X) : (Sing X)_n \to (Sing X)_{n-1}

that send each singular nn-simplex to its kk-face and functions

s kHom Top(σ k,X):(SingX) n(SingX) n+1 s_k \coloneqq Hom_{Top}(\sigma_k,X) : (Sing X)_{n} \to (Sing X)_{n+1}

that regard an nn-simplex as beign a degenerate (“thin”) (n+1)(n+1)-simplex. All these sets of simplices and face and degeneracy maps between them form the following structure.

Definition

(simplicial sets)

A simplicial set SS is

  • for each nn \in \mathbb{N} a set S nSetS_n \in Set – the set of nn-simplices;

  • for each injective map δ i:n1¯n¯\delta_i : \overline{n-1} \to \overline{n} of totally ordered sets n¯{0<1<<n}\bar n \coloneqq \{ 0 \lt 1 \lt \cdots \lt n \}

    a function d i:S nS n1d_i : S_{n} \to S_{n-1} – the iith face map on nn-simplices;

  • for each surjective map σ i:n+1¯n¯\sigma_i : \overline{n+1} \to \bar n of totally ordered sets

    a function σ i:S nS n+1\sigma_i : S_{n} \to S_{n+1} – the iith degeneracy map on nn-simplices;

such that these functions satisfy the following identities, called the_simplicial identities:

  1. d id j=d j1d i d_i \circ d_j = d_{j-1} \circ d_i if i<ji \lt j,

  2. s is j=s js i1s_i \circ s_j = s_j \circ s_{i-1} if i>ji \gt j.

  3. d is j={s j1d i ifi<j id ifi=jori=j+1 s jd i1 ifi>j+1d_i \circ s_j = \left\{ \array{ s_{j-1} \circ d_i & if \; i \lt j \\ id & if \; i = j \; or \; i = j+1 \\ s_j \circ d_{i-1} & if i \gt j+1 } \right.

For S,TS, T two simplicial sets, a morphism of simplicial sets SfTS \overset{f}{\longrightarrow} T is for each nn \in \mathbb{N} a function S Nf nT nS_N \overset{f_n}{\longrightarrow} T_n between sets of nn-simplices, such that these functions are compatible with all the face and degeneracy maps.

This defines a category (Def. ) sSet of simplicial sets.

It is straightforward to check by explicit inspection that the evident injection and restriction maps between the sets of singular simplices make (SingX) (Sing X)_\bullet into a simplicial set. However for working with this, it is good to streamline a little:

Definition

(simplex category)

The simplex category Δ\Delta is the full subcategory of Cat on the free categories of the form

[0] {0} [1] {01} [2] {012} . \begin{aligned} [0] & \coloneqq \{0\} \\ [1] & \coloneqq \{0 \to 1\} \\ [2] & \coloneqq \{0 \to 1 \to 2\} \\ \vdots \end{aligned} \,.
Remark

This is called the “simplex category” because we are to think of the object [n][n] as being the “spine” of the nn-simplex. For instance for n=2n = 2 we think of 0120 \to 1 \to 2 as the “spine” of the triangle. This becomes clear if we don’t just draw the morphisms that generate the category [n][n], but draw also all their composites. For instance for n=2n = 2 we have_

[2]={ 1 0 2}. [2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\} \,.
Proposition

(simplicial sets are presheaves on the simplex category)

A functor

S:Δ opSet S \;\colon\; \Delta^{op} \to Set

from the opposite category (Example ) of the simplex category (Def. ) to the category of sets, hence a presheaf on Δ\Delta (Example ), is canonically identified with a simplicial set, def. .

Via this identification, the category sSet of simplicial sets (Def. ) is equivalent to the category of presheaves on the simplex category

sSet=[Δ op,Set]. sSet \;=\; [\Delta^{op}, Set] \,.

In particular this means that sSet is a cosmos for enriched category theory (Example ), by Prop. .

Proof

One checks by inspection that the simplicial identities characterize precisely the behaviour of the morphisms in Δ op([n],[n+1])\Delta^{op}([n],[n+1]) and Δ op([n],[n1])\Delta^{op}([n],[n-1]).

This makes the following evident:

Example

The topological simplices from def. arrange into a cosimplicial object in Top, namely a functor

Δ :ΔTop. \Delta^\bullet : \Delta \to Top \,.

With this now the structure of a simplicial set on (SingX) (Sing X)_\bullet, def. , is manifest: it is just the nerve of XX with respect to Δ \Delta^\bullet, namely:

Definition

For XX a topological space its simplicial set of singular simplicies (often called the singular simplicial complex)

(SingX) :Δ opSet (Sing X)_\bullet : \Delta^{op} \to Set

is given by composition of the functor from example with the hom functor of Top:

(SingX):[n]Hom Top(Δ n,X). (Sing X) : [n] \mapsto Hom_{Top}( \Delta^n , X ) \,.
Remark

It turns out – this is the content of the homotopy hypothesis-theorem (Quillen 67) – that homotopy type of the topological space XX is entirely captured by its singular simplicial complex SingXSing X. Moreover, the geometric realization of SingXSing X is a model for the same homotopy type as that of XX, but with the special property that it is canonically a cell complex – a CW-complex. Better yet, SingXSing X is itself already good cell complex, namely a Kan complex. We come to this below.

Simplicial homotopy

The concept of homotopy of morphisms between simplicial sets proceeds in direct analogy with that in topological spaces.

Definition

For XX a simplicial set, def. , its simplicial cylinder object is the Cartesian product X×Δ[1]X\times \Delta[1] (formed in the category sSet, Prop. ).

A left homotopy

η:fg \eta \;\colon\; f \Rightarrow g

between two morphisms

f,g:XY f,g\;\colon\; X \longrightarrow Y

of simplicial sets is a morphism

η:X×Δ[1]Y \eta \;\colon\; X \times \Delta[1] \longrightarrow Y

such that the following diagram commutes

X (id X,d 1) f X×Δ 1 η Y (id x,d 0) g X. \array{ X \\ {}^{\mathllap{(id_X,d_1)}}\downarrow & \searrow^{\mathllap{f}} \\ X \times \Delta^1 &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id_x, d_0)}}\uparrow & \nearrow_{\mathllap{g}} \\ X } \,.

For YY a Kan complex, def. , its simplicial path space object is the function complex X Δ[1]X^{\Delta[1]} (formed in the category sSet, Prop. ).

A right homotopy

η:fg \eta \;\colon\; f \Rightarrow g

between two morphisms

f,g:XY f,g\;\colon\; X \longrightarrow Y

of simplicial sets is a morphism

η:XY Δ[1] \eta \colon X \longrightarrow Y^{\Delta[1]}

such that the following diagram commutes

Y f Y d 1 X η Y Δ[1] g Y d 0 Y. \array{ && Y \\ & {}^{\mathllap{f}}\nearrow & \uparrow^{\mathrlap{Y^{d_1}}} \\ X &\stackrel{\eta}{\longrightarrow}& Y^{\Delta[1]} \\ & {}_{\mathllap{g}}\searrow & \downarrow^{\mathrlap{Y^{d_0}}} \\ && Y } \,.
Proposition

For YY a Kan complex, def. , and XX any simplicial set, then left homotopy, def. , regarded as a relation

(fg)(fg) (f\sim g) \Leftrightarrow (f \stackrel{\exists}{\Rightarrow} g)

on the hom set Hom sSet(X,Y)Hom_{sSet}(X,Y), is an equivalence relation.

Definition

(homotopy equivalence in simplicial sets)

A morphism f:XYf \colon X \longrightarrow Y of simplicial sets is a left/right homotopy equivalence if there exists a morphisms XY:gX \longleftarrow Y \colon g and left/right homotopies (def. )

gfid X,fgid Y g \circ f \Rightarrow id_X\,,\;\;\;\; f\circ g \Rightarrow id_Y

The the basic invariants of simplicial sets/Kan complexes in simplicial homotopy theory are their simplicial homotopy groups, to which we turn now.

Given that a Kan complex is a special simplicial set that behaves like a combinatorial model for a topological space, the simplicial homotopy groups of a Kan complex are accordingly the combinatorial analog of the homotopy groups of topological spaces: instead of being maps from topological spheres modulo maps from topological disks, they are maps from the boundary of a simplex modulo those from the simplex itself.

Accordingly, the definition of the discussion of simplicial homotopy groups is essentially literally the same as that of homotopy groups of topological spaces. One technical difference is for instance that the definition of the group structure is slightly more non-immediate for simplicial homotopy groups than for topological homotopy groups (see below).

Definition

For XX a Kan complex, then its 0th simplicial homotopy group (or set of connected components) is the set of equivalence classes of vertices modulo the equivalence relation X 1(d 1,d 0)X 0×X 0X_1 \stackrel{(d_1,d_0)}{\longrightarrow} X_0 \times X_0

π 0(X):X 0/X 1. \pi_0(X) \colon X_0/X_1 \,.

For xX 0x \in X_0 a vertex and for nn \in \mathbb{N}, n1n \geq 1, then the underlying set of the nnth simplicial homotopy group of XX at xx – denoted π n(X,x)\pi_n(X,x) – is, the set of equivalence classes [α][\alpha] of morphisms

α:Δ nX \alpha \colon \Delta^n \to X

from the simplicial nn-simplex Δ n\Delta^n to XX, such that these take the boundary of the simplex to xx, i.e. such that they fit into a commuting diagram in sSet of the form

Δ[n] Δ[0] x Δ[n] α X, \array{ \partial \Delta[n] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] &\stackrel{\alpha}{\longrightarrow}& X } \,,

where two such maps α,α\alpha, \alpha' are taken to be equivalent is they are related by a simplicial homotopy η\eta

Δ[n] i 0 α Δ[n]×Δ[1] η X i 1 α Δ[n] \array{ \Delta[n] \\ \downarrow^{i_0} & \searrow^{\alpha} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X \\ \uparrow^{i_1} & \nearrow_{\alpha'} \\ \Delta[n] }

that fixes the boundary in that it fits into a commuting diagram in sSet of the form

Δ[n]×Δ[1] Δ[0] x Δ[n]×Δ[1] η X. \array{ \partial \Delta[n] \times \Delta[1] & \longrightarrow & \Delta[0] \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Delta[n] \times \Delta[1] &\stackrel{\eta}{\longrightarrow}& X } \,.

These sets are taken to be equipped with the following group structure.

Definition

For XX a Kan complex, for xX 0x\in X_0, for n1n \geq 1 and for f,g:Δ[n]Xf,g \colon \Delta[n] \to X two representatives of π n(X,x)\pi_n(X,x) as in def. , consider the following nn-simplices in X nX_n:

v i{s 0s 0s 0(x) for0in2 f fori=n1 g fori=n+1 v_i \coloneqq \left\{ \array{ s_0 \circ s_0 \circ \cdots \circ s_0 (x) & for \; 0 \leq i \leq n-2 \\ f & for \; i = n-1 \\ g & for \; i = n+1 } \right.

This corresponds to a morphism Λ n+1[n]X\Lambda^{n+1}[n] \to X from a horn of the (n+1)(n+1)-simplex into XX. By the Kan complex property of XX this morphism has an extension θ\theta through the (n+1)(n+1)-simplex Δ[n]\Delta[n]

Λ n+1[n] X θ Δ[n+1] \array{ \Lambda^{n+1}[n] & \longrightarrow & X \\ \downarrow & \nearrow_{\mathrlap{\theta}} \\ \Delta[n+1] }

From the simplicial identities one finds that the boundary of the nn-simplex arising as the nnth boundary piece d nθd_n \theta of θ\theta is constant on xx

d id nθ=d n1d iθ=x d_i d_{n} \theta = d_{n-1} d_i \theta = x

So d nθd_n \theta represents an element in π n(X,x)\pi_n(X,x) and we define a product operation on π n(X,x)\pi_n(X,x) by

[f][g][d nθ]. [f]\cdot [g] \coloneqq [d_n \theta] \,.

(e.g. Goerss-Jardine 99, p. 26)

Remark

All the degenerate nn-simplices v 0in2v_{0 \leq i \leq n-2} in def. are just there so that the gluing of the two nn-cells ff and gg to each other can be regarded as forming the boundary of an (n+1)(n+1)-simplex except for one face. By the Kan extension property that missing face exists, namely d nθd_n \theta. This is a choice of gluing composite of ff with gg.

Lemma

The product on homotopy group elements in def. is well defined, in that it is independent of the choice of representatives ff, gg and of the extension θ\theta.

e.g. (Goerss-Jardine 99, lemma 7.1)

Lemma

The product operation in def. yields a group structure on π n(X,x)\pi_n(X,x), which is abelian for n2n \geq 2.

e.g. (Goerss-Jardine 99, theorem 7.2)

Remark

The first homotopy group, π 1(X,x)\pi_1(X,x), is also called the fundamental group of XX.

Definition

(weak homotopy equivalence of simplicial sets)

For X,YKanCplxsSetX,Y \in KanCplx \hookrightarrow sSet two Kan complexes, then a morphism

f:XY f \colon X \longrightarrow Y

is called a weak homotopy equivalence if it induces isomorphisms on all simplicial homotopy groups, i.e. if

  1. π 0(f):π 0(X)π 0(Y)\pi_0(f) \colon \pi_0(X) \longrightarrow \pi_0(Y) is a bijection of sets;

  2. π n(f,x):π n(X,x)π n(Y,f(x))\pi_n(f,x) \colon \pi_n(X,x) \longrightarrow \pi_n(Y,f(x)) is an isomorphism of groups for all xX 0x\in X_0 and all nn \in \mathbb{N}; n1n \geq 1.

Kan complexes

Recall the definition of simplicial sets from above. Let

Δ[n]=Δ(,[n])SimpSet \Delta[n] = \mathbf{\Delta}( -, [n]) \in Simp Set

be the standard simplicial nn-simplex in SimpSet.

Definition

For each ii, 0in0 \leq i \leq n, the (n,i)(n,i)-horn or (n,i)(n,i)-box is the subsimplicial set

Λ i[n]Δ[n] \Lambda^i[n] \hookrightarrow \Delta[n]

which is the union of all faces except the i thi^{th} one.

This is called an outer horn if k=0k = 0 or k=nk = n. Otherwise it is an inner horn.

Remark

Since sSet is a presheaf topos, unions of subobjects make sense and they are calculated objectwise, thus in this case dimensionwise. This way it becomes clear what the structure of a horn as a functor Λ k[n]:Δ opSet\Lambda^k[n]: \Delta^{op} \to Set must therefore be: it takes [m][m] to the collection of ordinal maps f:[m][n]f: [m] \to [n] which do not have the element kk in the image.

Example

The inner horn, def. of the 2-simplex

Δ 2={ 1 0 2} \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \right\}

with boundary

Δ 2={ 1 0 2}\partial \Delta^2 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&\to&& 2 } \right\}

looks like

Λ 1 2={ 1 0 2}. \Lambda^2_1 = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\} \,.

The two outer horns look like

Λ 0 2={ 1 0 2}\Lambda^2_0 = \left\{ \array{ && 1 \\ & \nearrow && \\ 0 &&\to&& 2 } \right\}

and

Λ 2 2={ 1 0 2}\Lambda^2_2 = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

respectively.

(graphics taken from Friedman 08)

Definition

(Kan complex)

A Kan complex is a simplicial set SS that satisfies the Kan condition,

  • which says that all horns of the simplicial set have fillers/extend to simplices;

  • which means equivalently that the unique homomorphism SptS \to pt from SS to the point (the terminal simplicial set) is a Kan fibration;

  • which means equivalently that for all diagrams in sSet of the form

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }

    there exists a diagonal morphism

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] }

    completing this to a commuting diagram;

  • which in turn means equivalently that the map from nn-simplices to (n,i)(n,i)-horns is an epimorphism

    [Δ[n],S][Λ i[n],S]. [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,.
Proposition

For XX a topological space, its singular simplicial complex Sing(X)Sing(X), def. , is a Kan complex, def. .

Proof

The inclusions Λ n Top kΔ Top n{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top} of topological horns into topological simplices are retracts, in that there are continuous maps Δ Top nΛ n Top k\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k given by “squashing” a topological nn-simplex onto parts of its boundary, such that

(Λ n Top kΔ Top nΛ n Top k)=Id. ({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,.

Therefore the map [Δ n,Π(X)][Λ k n,Π(X)][\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)] is an epimorphism, since it is equal to to Top(Δ n,X)Top(Λ k n,X)Top(\Delta^n, X) \to Top(\Lambda^n_k, X) which has a right inverse Top(Λ k n,X)Top(Δ n,X)Top(\Lambda^n_k, X) \to Top(\Delta^n, X).

More generally:

Definition

(Kan fibration)

A morphism ϕ:ST\phi \colon S \longrightarrow T in sSet is called a Kan fibration if it has the right lifting property again all horn inclusions, def. , hence if for every commuting diagram of the form

Λ i[n] S ϕ Δ[n] T \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T }

there exists a lift

Λ i[n] S ϕ Δ[n] T. \array{ \Lambda^i[n] &\longrightarrow& S \\ \downarrow &\nearrow& \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T } \,.

This is the simplicial incarnation of the concept of Serre fibrations of topological spaces:

Definition

A continuous function f:XYf \colon X \longrightarrow Y between topological spaces is a Serre fibration if for all CW-complexes CC and for every commuting diagram in Top of the form

C X f C×I Y \array{ C &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y }

there exists a lift

C X f C×I Y. \array{ C &\longrightarrow& X \\ \downarrow &\nearrow& \downarrow^{\mathrlap{f}} \\ C \times I &\longrightarrow& Y } \,.
Proposition

A continuous function f:XYf \colon X \longrightarrow Y is a Serre fibration, def. , precisely if Sing(f):Sing(X)Sing(Y)Sing(f) \colon Sing(X) \longrightarrow Sing(Y) (def. ) is a Kan fibration, def. .

The proof uses the basic tool of nerve and realization-adjunction to which we get to below in prop. .

Proof

First observe that the left lifting property against all CC×IC \hookrightarrow C \times I for CC a CW-complex is equivalent to left lifting against geometric realization |Λ i[n]||Δ[n]|{\vert \Lambda^i[n]\vert} \hookrightarrow {\vert \Delta[n]\vert} of horn inclusions. Then apply the natural isomorphism Top(||,)sSet(,Sing())Top({\vert-\vert},-) \simeq sSet(-,Sing(-)), given by the adjunction of prop. and example , to the lifting diagrams.

Lemma

Let p:XYp \colon X \longrightarrow Y be a Kan fibration, def. , and let f 1,f 2:AXf_1,f_2 \colon A \longrightarrow X be two morphisms. If there is a left homotopy (def. ) f 1f 2f_1 \Rightarrow f_2 between these maps, then there is a fiberwise homotopy equivalence, def. , between the pullback fibrations f 1 *Xf 2 *Xf_1^\ast X \simeq f_2^\ast X.

(e.g. Goerss-Jardine 99, chapter I, lemma 10.6)

While simplicial sets have the advantage of being purely combinatorial structures, the singular simplicial complex of any given topological space, def. is in general a huge simplicial set which does not lend itself to detailed inspection. The following is about small models.

Definition

A Kan fibration ϕ:ST\phi \colon S \longrightarrow T, def. , is called a minimal Kan fibration if for any two cells in the same fiber with the same boundary if they are homotopic relative their boundary, then they are already equal.

More formally, ϕ\phi is minimal precisely if for every commuting diagram of the form

(Δ[n])×Δ[1] p 1 Δ[n] Δ[n]×Δ[1] h S p 1 ϕ Δ[n] T \array{ (\partial \Delta[n]) \times \Delta[1] &\stackrel{p_1}{\longrightarrow}& \partial \Delta[n] \\ \downarrow && \downarrow \\ \Delta[n] \times \Delta[1] &\stackrel{h}{\longrightarrow}& S \\ \downarrow^{\mathrlap{p_1}} && \downarrow^{\mathrlap{\phi}} \\ \Delta[n] &\longrightarrow& T }

then the two composites

Δ[n]d 1d 0Δ[n]×Δ[1]hS \Delta[n] \stackrel{\overset{d_0}{\longrightarrow}}{\underset{d_1}{\longrightarrow}} \Delta[n] \times \Delta[1] \stackrel{h}{\longrightarrow} S

are equal.

Proposition

The pullback (in sSet) of a minimal Kan fibration, def. , along any morphism is again a mimimal Kan fibration.

anodyne extensions

(Goerss-Jardine 99, chapter I, section 4, Joyal-Tierney 05, section 31)

Proposition

For every Kan fibration, def. , there exists a fiberwise strong deformation retract to a minimal Kan fibration, def. .

(e.g. Goerss-Jardine 99, chapter I, prop. 10.3, Joyal-Tierney 05, theorem 3.3.1, theorem 3.3.3).

Proof idea

Choose representatives by induction, use that in the induction step one needs lifts of anodyne extensions against a Kan fibration, which exist.

Lemma

A morphism between minimal Kan fibrations, def. , which is fiberwise a homotopy equivalence, def. , is already an isomorphism.

(e.g. Goerss-Jardine 99, chapter I, lemma 10.4)

Proof idea

Show the statement degreewise. In the induction one needs to lift anodyne extensions agains a Kan fibration.

Lemma

Every minimal Kan fibration, def. , over a connected base is a simplicial fiber bundle, locally trivial over every simplex of the base.

(e.g. Goerss-Jardine 99, chapter I, corollary 10.8)

Proof

By assumption of the base being connected, the classifying maps for the fibers over any two vertices are connected by a zig-zag of homotopies, hence by lemma the fibers are connected by homotopy equivalences and then by prop. and lemma they are already isomorphic. Write FF for this typical fiber.

Moreover, for all nn the morphisms Δ[n]Δ[0]Δ[n]\Delta[n] \to \Delta[0] \to \Delta[n] are left homotopic to Δ[n]idΔ[n]\Delta[n] \stackrel{id}{\to} \Delta[n] and so applying lemma and prop. once more yields that the fiber over each Δ[n]\Delta[n] is isomorphic to Δ[n]×F\Delta[n]\times F.

Groupoids as Kan complexes

Definition

A (small) groupoid 𝒢 \mathcal{G}_\bullet is

  • a pair of sets 𝒢 0Set\mathcal{G}_0 \in Set (the set of objects) and 𝒢 1Set\mathcal{G}_1 \in Set (the set of morphisms)

  • equipped with functions

    𝒢 1× 𝒢 0𝒢 1 𝒢 1 sit 𝒢 0, \array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\longrightarrow}& \mathcal{G}_1 & \stackrel{\overset{t}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\longrightarrow}}}& \mathcal{G}_0 }\,,

    where the fiber product on the left is that over 𝒢 1t𝒢 0s𝒢 1\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1,

such that

  • ii takes values in endomorphisms;

    ti=si=id 𝒢 0, t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;
  • \circ defines a partial composition operation which is associative and unital for i(𝒢 0)i(\mathcal{G}_0) the identities; in particular

    s(gf)=s(f)s (g \circ f) = s(f) and t(gf)=t(g)t (g \circ f) = t(g);

  • every morphism has an inverse under this composition.

Remark

This data is visualized as follows. The set of morphisms is

𝒢 1={ϕ 0kϕ 1} \mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\}

and the set of pairs of composable morphisms is

𝒢 2𝒢 1×𝒢 0𝒢 1={ ϕ 1 k 1 k 2 ϕ 0 k 2k 1 ϕ 2}. \mathcal{G}_2 \coloneqq \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 = \left\{ \array{ && \phi_1 \\ & {}^{\mathllap{k_1}}\nearrow && \searrow^{\mathrlap{k_2}} \\ \phi_0 && \stackrel{k_2 \circ k_1}{\to} && \phi_2 } \right\} \,.

The functions p 1,p 2,:𝒢 2𝒢 1p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1 are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.

Example

For XX a set, it becomes a groupoid by taking XX to be the set of objects and adding only precisely the identity morphism from each object to itself

(XidididX). \left( X \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longrightarrow}} { \overset{id}{\longleftarrow} } X \right) \,.
Example

For GG a group, its delooping groupoid (BG) (\mathbf{B}G)_\bullet has

  • (BG) 0=*(\mathbf{B}G)_0 = \ast;

  • (BG) 1=G(\mathbf{B}G)_1 = G.

For GG and KK two groups, group homomorphisms f:GKf \colon G \to K are in natural bijection with groupoid homomorphisms

(Bf) :(BG) (BK) . (\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,.

In particular a group character c:GU(1)c \colon G \to U(1) is equivalently a groupoid homomorphism

(Bc) :(BG) (BU(1)) . (\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,.

Here, for the time being, all groups are discrete groups. Since the circle group U(1)U(1) also has a standard structure of a Lie group, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on

U(1)Grp \flat U(1) \in Grp

to mean explicitly the discrete group underlying the circle group. (Here “\flat” denotes the “flat modality”.)

Example

For XX a set, GG a discrete group and ρ:X×GX\rho \colon X \times G \to X an action of GG on XX (a permutation representation), the action groupoid or homotopy quotient of XX by GG is the groupoid

X// ρG=(X×Gp 1ρX) X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right)

with composition induced by the product in GG. Hence this is the groupoid whose objects are the elements of XX, and where morphisms are of the form

x 1gx 2=ρ(x 1)(g) x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g)

for x 1,x 2Xx_1, x_2 \in X, gGg \in G.

As an important special case we have:

Example

For GG a discrete group and ρ\rho the trivial action of GG on the point *\ast (the singleton set), the corresponding action groupoid according to def. is the delooping groupoid of GG according to def. :

(*//G) =(BG) . (\ast //G)_\bullet = (\mathbf{B}G)_\bullet \,.

Another canonical action is the action of GG on itself by right multiplication. The corresponding action groupoid we write

(EG) G//G. (\mathbf{E}G)_\bullet \coloneqq G//G \,.

The constant map G*G \to \ast induces a canonical morphism

G//G EG *//G BG. \array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,.

This is known as the GG-universal principal bundle. See below in for more on this.

Example

The interval II is the groupoid with

  • I 0={a,b}I_0 = \{a,b\};
  • I 1={id a,id b,ab}I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}.
Example

For Σ\Sigma a topological space, its fundamental groupoid Π 1(Σ)\Pi_1(\Sigma) is

  • Π 1(Σ) 0=\Pi_1(\Sigma)_0 = points in XX;
  • Π 1(Σ) 1=\Pi_1(\Sigma)_1 = continuous paths in XX modulo homotopy that leaves the endpoints fixed.
Example

For 𝒢 \mathcal{G}_\bullet any groupoid, there is the path space groupoid 𝒢 I\mathcal{G}^I_\bullet with

  • 𝒢 0 I=𝒢 1={ϕ 0 k ϕ 1}\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\};

  • 𝒢 1 I=\mathcal{G}^I_1 = commuting squares in 𝒢 \mathcal{G}_\bullet = {ϕ 0 h 0 ϕ˜ 0 k k˜ ϕ 1 h 1 ϕ˜ 1}. \left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,.

This comes with two canonical homomorphisms

𝒢 Iev 0ev 1𝒢 \mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet

which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.

Definition

For f ,g :𝒢 𝒦 f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet two morphisms between groupoids, a homotopy fgf \Rightarrow g (a natural transformation) is a homomorphism of the form η :𝒢 𝒦 I\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet (with codomain the path space object of 𝒦 \mathcal{K}_\bullet as in example ) such that it fits into the diagram as depicted here on the right:

f 𝒢 η 𝒦 g 𝒦 f (ev 1) 𝒢 η 𝒦 I g (ev 0) 𝒦. \array{ & \nearrow \searrow^{\mathrlap{f_\bullet}} \\ \mathcal{G} &\Downarrow^{\mathrlap{\eta}}& \mathcal{K} \\ & \searrow \nearrow_{\mathrlap{g_\bullet}} } \;\;\;\; \coloneqq \;\;\;\; \array{ && \mathcal{K}_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{(ev_1)_\bullet}} \\ \mathcal{G}_\bullet &\stackrel{\eta_\bullet}{\to}& \mathcal{K}^I_\bullet \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow^{\mathrlap{(ev_0)_\bullet}} \\ && \mathcal{K} } \,.
Definition (Notation)

Here and in the following, the convention is that we write

  • 𝒢 \mathcal{G}_\bullet (with the subscript decoration) when we regard groupoids with just homomorphisms (functors) between them,

  • 𝒢\mathcal{G} (without the subscript decoration) when we regard groupoids with homomorphisms (functors) between them and homotopies (natural transformations) between these

    f X Y g. \array{ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{g} } \,.

The unbulleted version of groupoids are also called homotopy 1-types (or often just their homotopy-equivalence classes are called this way.) Below we generalize this to arbitrary homotopy types (def. ).

Example

For X,YX,Y two groupoids, the mapping groupoid [X,Y][X,Y] or Y XY^X is

  • [X,Y] 0=[X,Y]_0 = homomorphisms XYX \to Y;
  • [X,Y] 1=[X,Y]_1 = homotopies between such.
Definition

A (homotopy-) equivalence of groupoids is a morphism 𝒢𝒦\mathcal{G} \to \mathcal{K} which has a left and right inverse up to homotopy.

Example

The map

BΠ(S 1) \mathbf{B}\mathbb{Z} \stackrel{}{\to} \Pi(S^1)

which picks any point and sends nn \in \mathbb{Z} to the loop based at that point which winds around nn times, is an equivalence of groupoids.

Proposition

Assuming the axiom of choice in the ambient set theory, every groupoid is equivalent to a disjoint union of delooping groupoids, example – a skeleton.

Remark

The statement of prop. becomes false as when we pass to groupoids that are equipped with geometric structure. This is the reason why for discrete geometry all Chern-Simons-type field theories (namely Dijkgraaf-Witten theory-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the AKSZ sigma-models. But even so, Dijkgraaf-Witten theory is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. is not canonical.

Definition

Given two morphisms of groupoids XfBgYX \stackrel{f}{\rightarrow} B \stackrel{g}{\leftarrow} Y their homotopy fiber product

X×BY X f Y g B \array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B }

is the limit cone

X ×B B I×B Y X f B I (ev 0) B (ev 1) Y g B , \array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,,

hence the ordinary iterated fiber product over the path space groupoid, as indicated.

Remark

An ordinary fiber product X ×B Y X_\bullet \underset{B_\bullet}{\times}Y_\bullet of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:

(X ×B Y ) i=X i×B iY i. (X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,.
Example

For XX a groupoid, GG a group and XBGX \to \mathbf{B}G a map into its delooping, the pullback PXP \to X of the GG-universal principal bundle of example is equivalently the homotopy fiber product of XX with the point over BG\mathbf{B}G:

PX×BG*. P \simeq X \underset{\mathbf{B}G}{\times} \ast \,.

Namely both squares in the following diagram are pullback squares

P EG * (BG) I (ev 0) (BG) (ev 1) X (BG) . \array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,.

(This is the first example of the more general phenomenon of universal principal infinity-bundles.)

Example

For XX a groupoid and *X\ast \to X a point in it, we call

ΩX*×X* \Omega X \coloneqq \ast \underset{X}{\times} \ast

the loop space groupoid of XX.

For GG a group and BG\mathbf{B}G its delooping groupoid from example , we have

GΩBG=*×BG*. G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,.

Hence GG is the loop space object of its own delooping, as it should be.

Proof

We are to compute the ordinary limiting cone *×BG (BG I) ×BG *\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast in

* (BG) I (ev 0) BG (ev 1) * BG , \array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,,

In the middle we have the groupoid (BG) I(\mathbf{B}G)^I_\bullet whose objects are elements of GG and whose morphisms starting at some element are labeled by pairs of elements h 1,h 2Gh_1, h_2 \in G and end at h 1gh 2h_1 \cdot g \cdot h_2. Using remark the limiting cone is seen to precisely pick those morphisms in (BG ) I(\mathbf{B}G_\bullet)^I_\bullet such that these two elements are constant on the neutral element h 1=h 2=e=id *h_1 = h_2 = e = id_{\ast}, hence it produces just the elements of GG regarded as a groupoid with only identity morphisms, as in example .

Proposition

The free loop space object is

[Π(S 1),X]X×[Π(S 0),X]X [\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X
Proof

Notice that Π 1(S 0)**\Pi_1(S^0) \simeq \ast \coprod \ast. Therefore the path space object [Π(S 0),X ] I[\Pi(S^0), X_\bullet]^I_\bullet has

  • objects are pairs of morphisms in X X_\bullet;

  • morphisms are commuting squares of such.

Now the fiber product in def. picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore X ×[Π(S 0),X ] [Π(S 0),X ] I×[Π(S 0),X ] XX_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X is the groupoid whose

  • objects are diagrams in X X_\bullet of the form

    x 0 x 1 \array{ & \nearrow \searrow \\ x_0 && x_1 \\ & \searrow \nearrow }
  • morphism are cylinder-diagrams over these.

One finds along the lines of example that this is equivalent to maps from Π 1(S 1)\Pi_1(S^1) into X X_\bullet and homotopies between these.

Remark

Even though all these models of the circle Π 1(S 1)\Pi_1(S^1) are equivalent, below the special appearance of the circle in the proof of prop. as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an evaluation map with a coevaluation map.

Example

For GG a discrete group, the free loop space object of its delooping BG\mathbf{B}G is G// adGG//_{ad} G, the action groupoid, def. , of the adjoint action of GG on itself:

[Π(S 1),BG]G// adG. [\Pi(S^1), \mathbf{B}G] \simeq G//_{ad} G \,.
Example

For an abelian group such as U(1)\flat U(1) we have

[Π(S 1),BU(1)]U(1)// adU(1)(U(1))×(BU(1)). [\Pi(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,.
Example

Let c:GU(1)c \colon G \to \flat U(1) be a group homomorphism, hence a group character. By example this has a delooping to a groupoid homomorphism

Bc:BGBU(1). \mathbf{B}c \;\colon\; \mathbf{B}G \to \mathbf{B}\flat U(1) \,.

Under the free loop space object construction this becomes

[Π(S 1),Bc]:[Π(S 1),BG][Π(S 1),BU(1)] [\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)]

hence

[Π(S 1),Bc]:G// adGU(1)×BU(1). [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,.

So by postcomposing with the projection on the first factor we recover from the general homotopy theory of groupoids the statement that a group character is a class function on conjugacy classes:

[Π(S 1),Bc]:G// adGU(1). [\Pi(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to U(1) \,.

\,

Definition

For 𝒢 \mathcal{G}_\bullet a groupoid, def. , its simplicial nerve N(𝒢 ) N(\mathcal{G}_\bullet)_\bullet is the simplicial set with

N(𝒢 ) n𝒢 1 × 𝒢 0 n N(\mathcal{G}_\bullet)_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n}

the set of sequences of composable morphisms of length nn, for nn \in \mathbb{N};

with face maps

d k:N(𝒢 ) n+1N(𝒢 ) n d_k \colon N(\mathcal{G}_\bullet)_{n+1} \to N(\mathcal{G}_\bullet)_{n}

being,

  • for n=0n = 0 the functions that remembers the kkth object;

  • for n1n \geq 1

    • the two outer face maps d 0d_0 and d nd_n are given by forgetting the first and the last morphism in such a sequence, respectively;

    • the n1n-1 inner face maps d 0<k<nd_{0 \lt k \lt n} are given by composing the kkth morphism with the k+1k+1st in the sequence.

The degeneracy maps

s k:N(𝒢 )nN(𝒢 ) n+1. s_k \colon N(\mathcal{G}_\bullet)n \to N(\mathcal{G}_\bullet)_{n+1} \,.

are given by inserting an identity morphism on x kx_k.

Remark

Spelling this out in more detail: write

𝒢 n={x 0f 0,1x 1f 1,2x 2f 2,3f n1,nx n} \mathcal{G}_n = \left\{ x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_2 \stackrel{f_{2,3}}{\to} \cdots \stackrel{f_{n-1,n}}{\to} x_n \right\}

for the set of sequences of nn composable morphisms. Given any element of this set and 0<k<n0 \lt k \lt n , write

f i1,i+1f i,i+1f i1,i f_{i-1,i+1} \coloneqq f_{i,i+1} \circ f_{i-1,i}

for the comosition of the two morphism that share the iith vertex.

With this, face map d kd_k acts simply by “removing the index kk”:

d 0:(x 0f 0,1x 1f 1,2x 2f n1,nx n)(x 1f 1,2x 2f n1,nx n) d_0 \colon (x_0 \stackrel{f_{0,1}}{\to} x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto (x_1 \stackrel{f_{1,2}}{\to} x_{2} \cdots \stackrel{f_{n-1,n}}{\to} x_n )
d 0<k<n:(x 0x k1f k1,kx kf k,k+1x k+1x n)(x 0x k1f k1,k+1x k+1x n) d_{0\lt k \lt n} \colon ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n ) \mapsto ( x_0 \cdots \stackrel{}{\to} x_{k-1} \stackrel{f_{k-1,k+1}}{\to} x_{k+1} \stackrel{}{\to} \cdots x_n )
d n:(x 0f 0,1f n2,n1x n1f n1,nx n)(x 0f 0,1f n2,n1x n1). d_n \colon ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} \stackrel{f_{n-1,n}}{\to} x_n ) \mapsto ( x_0 \stackrel{f_{0,1}}{\to} \cdots \stackrel{f_{n-2,n-1}}{\to} x_{n-1} ) \,.

Similarly, writing

f k,kid x k f_{k,k} \coloneqq id_{x_k}

for the identity morphism on the object x kx_k, then the degenarcy map acts by “repeating the kkth index”

s k:(x 0x kf k,k+1x k+1)(x 0x kf k,kx kf k,k+1x k+1). s_k \colon ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \mapsto ( x_0 \stackrel{}{\to} \cdots \to x_k \stackrel{f_{k,k}}{\to} x_k \stackrel{f_{k,k+1}}{\to} x_{k+1} \to \cdots ) \,.

This makes it manifest that these functions organise into a simplicial set.

Proposition

These collections of maps in def. satisfy the simplicial identities, hence make the nerve 𝒢 \mathcal{G}_\bullet into a simplicial set. Moreover, this simplicial set is a Kan complex, where each horn has a unique filler (extension to a simplex).

(A 2-coskeletal Kan complex.)

Proposition

The nerve operation constitutes a full and faithful functor

N:GrpdKanCplxsSet. N \colon Grpd \to KanCplx \hookrightarrow sSet \,.

Chain complexes as Kan complexes

In the familiar construction of singular homology recalled above one constructs the alternating face map chain complex of the simplicial abelian group of singular simplices, def. . This construction is natural and straightforward, but the result chain complex tends to be very “large” even if its chain homology groups end up being very “small”. But in the context of homotopy theory one is to consider all objects notup to isomorphism, but of to weak equivalence, which for chain complexes means up to quasi-isomorphisms. Hence one should look for the natural construction of “smaller” chain complexes that are still quasi-isomorphic to these alternating face map complexes. This is accomplished by the normalized chain complex construction:

Definition

For AA a simplicial abelian group its alternating face map complex (CA) (C A)_\bullet of AA is the chain complex which

  • in degree nn is given by the group A nA_n itself

    (CA) n:=A n (C A)_n := A_n
  • with differential given by the alternating sum of face maps (using the abelian group structure on AA)

    n i=0 n(1) id i:(CA) n(CA) n1. \partial_n \coloneqq \sum_{i = 0}^n (-1)^i d_i \;\colon\; (C A)_n \to (C A)_{n-1} \,.
Lemma

The differential in def. is well-defined in that it indeed squares to 0.

Proof

Using the simplicial identity, prop. , d id j=d j1d id_i \circ d_j = d_{j-1} \circ d_i for i<ji \lt j one finds:

n n+1 = i,j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd id j = ij(1) i+jd id j+ i<j(1) i+jd j1d i = ij(1) i+jd id j ik(1) i+kd kd i =0. \begin{aligned} \partial_n \partial_{n+1} & = \sum_{i, j} (-1)^{i+j} d_i \circ d_{j} \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_i \circ d_j \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j + \sum_{i \lt j} (-1)^{i+j} d_{j-1} \circ d_i \\ &= \sum_{i \geq j} (-1)^{i+j} d_i \circ d_j - \sum_{i \leq k} (-1)^{i+k} d_{k} \circ d_i \\ &= 0 \end{aligned} \,.
Definition

Given a simplicial abelian group AA, its normalized chain complex or Moore complex is the \mathbb{N}-graded chain complex ((NA) ,)((N A)_\bullet,\partial ) which

  • is in degree nn the joint kernel

    (NA) n= i=1 nkerd i n (N A)_n=\bigcap_{i=1}^{n}ker\,d_i^n

    of all face maps except the 0-face;

  • with differential given by the remaining 0-face map

    n:=d 0 n| (NA) n:(NA) n(NA) n1. \partial_n := d_0^n|_{(N A)_n} : (N A)_n \rightarrow (N A)_{n-1} \,.
Remark

We may think of the elements of the complex NAN A, def. , in degree kk as being kk-dimensional disks in AA all whose boundary is captured by a single face:

  • an element gNG 1g \in N G_1 in degree 1 is a 1-disk

    1gg, 1 \stackrel{g}{\to} \partial g \,,
  • an element hNG 2h \in N G_2 is a 2-disk

    1 1 h h 1 1 1, \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,
  • a degree 2 element in the kernel of the boundary map is such a 2-disk that is closed to a 2-sphere

    1 1 h h=1 1 1 1, \array{ && 1 \\ & {}^1\nearrow &\Downarrow^h& \searrow^{\partial h = 1} \\ 1 &&\stackrel{1}{\to}&& 1 } \,,

etc.

Definition

Given a simplicial group AA (or in fact any simplicial set), then an element aA n+1a \in A_{n+1} is called degenerate (or thin) if it is in the image of one of the simplicial degeneracy maps s i:A nA n+1s_i \colon A_n \to A_{n+1}. All elements of A 0A_0 are regarded a non-degenerate. Write

D(A n+1) is i(A n)A n+1 D (A_{n+1}) \coloneqq \langle \cup_i s_i(A_{n}) \rangle \hookrightarrow A_{n+1}

for the subgroup of A n+1A_{n+1} which is generated by the degenerate elements (i.e. the smallest subgroup containing all the degenerate elements).

Definition

For AA a simplicial abelian group its alternating face maps chain complex modulo degeneracies, (CA)/(DA)(C A)/(D A) is the chain complex

  • which in degree 0 equals is just ((CA)/D(A)) 0A 0((C A)/D(A))_0 \coloneqq A_0;

  • which in degree n+1n+1 is the quotient group obtained by dividing out the group of degenerate elements, def. :

    ((CA)/D(A)) n+1:=A n+1/D(A n+1) ((C A)/D(A))_{n+1} := A_{n+1} / D(A_{n+1})
  • whose differential is the induced action of the alternating sum of faces on the quotient (which is well-defined by lemma ).

Lemma

Def. is indeed well defined in that the alternating face map differential respects the degenerate subcomplex.

Proof

Using the mixed simplicial identities we find that for s j(a)A ns_j(a) \in A_n a degenerate element, its boundary is

i(1) id is j(a) = i<j(1) is j1d i(a)+ i=j,j+1(1) ia+ i>j+1(1) is jd i1(a) = i<j(1) is j1d i(a)+ i>j+1(1) is jd i1(a) \begin{aligned} \sum_i (-1)^i d_i s_j(a) &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i = j, j+1} (-1)^i a + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \\ &= \sum_{i \lt j} (-1)^i s_{j-1} d_i(a) + \sum_{i \gt j+1} (-1)^i s_j d_{i-1}(a) \end{aligned}

which is again a combination of elements in the image of the degeneracy maps.

Proposition

Given a simplicial abelian group AA, the evident composite of natural morphisms

NAiAp(CA)/(DA) N A \stackrel{i}{\hookrightarrow} A \stackrel{p}{\to} (C A)/(D A)

from the normalized chain complex, def. , into the alternating face map complex modulo degeneracies, def. , (inclusion followed by projection to the quotient) is a natural isomorphism of chain complexes.

e.g. (Goerss-Jardine, theorem III 2.1).

Corollary

For AA a simplicial abelian group, there is a splitting

C (A)N (A)D (A) C_\bullet(A) \simeq N_\bullet(A) \oplus D_\bullet(A)

of the alternating face map complex, def. as a direct sum, where the first direct summand is naturally isomorphic to the normalized chain complex of def. and the second is the degenerate cells from def. .

Proof

By prop. there is an inverse to the diagonal composite in

CA p (CA)/(DA) i NA. \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A } \,.

This hence exhibits a splitting of the short exact sequence given by the quotient by DAD A.

0 DA CA p (CA)/(DA) 0 i iso NA. \array{ 0 &\to& D A &\hookrightarrow & C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) &\to & 0 \\ && && {}^{\mathllap{i}}\uparrow & \swarrow_{\mathrlap{\simeq}_{iso}} \\ && && N A } \,.
Theorem (Eilenberg-MacLane)

Given a simplicial abelian group AA, then the inclusion

NACA N A \hookrightarrow C A

of the normalized chain complex, def. into the full alternating face map complex, def. , is a natural quasi-isomorphism and in fact a natural chain homotopy equivalence, i.e. the complex D (X)D_\bullet(X) is null-homotopic.

(Goerss-Jardine, theorem III 2.4)

Corollary

Given a simplicial abelian group AA, then the projection chain map

(CA)(CA)/(DA) (C A) \longrightarrow (C A)/(D A)

from its alternating face maps complex, def. , to the alternating face map complex modulo degeneracies, def. , is a quasi-isomorphism.

Proof

Consider the pre-composition of the map with the inclusion of the normalized chain complex, def. .

CA p (CA)/(DA) i NA \array{ C A &\stackrel{p}{\longrightarrow}& (C A)/(D A) \\ {}^{\mathllap{i}}\uparrow & \nearrow \\ N A }

By theorem the vertical map is a quasi-isomorphism and by prop. the composite diagonal map is an isomorphism, hence in particular also a quasi-isomorphism. Since quasi-isomorphisms satisfy the two-out-of-three property, it follows that also the map in question is a quasi-isomorphism.

Example

Consider the 1-simplex Δ[1]\Delta[1] regarded as a simplicial set, and write [Δ[1]]\mathbb{Z}[\Delta[1]] for the simplicial abelian group which in each degree is the free abelian group on the simplices in Δ[1]\Delta[1].

This simplicial abelian group starts out as

[Δ[1]]=( 4 3 1 0 2) \mathbb{Z}[\Delta[1]] = \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} \mathbb{Z}^4 \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} \mathbb{Z}^3 \stackrel{\overset{\partial_0}{\longrightarrow}}{\underset{\partial_1}{\longrightarrow}} \mathbb{Z}^2 \right)

(where we are indicating only the face maps for notational simplicity).

Here the first 2=\mathbb{Z}^2 = \mathbb{Z}\oplus \mathbb{Z}, the direct sum of two copies of the integers, is the group of 0-chains generated from the two endpoints (0)(0) and (1)(1) of Δ[1]\Delta[1], i.e. the abelian group of formal linear combinations of the form

2{a(0)+b(1)|a,b}. \mathbb{Z}^2 \simeq \left\{ a \cdot (0) + b \cdot (1) | a,b \in \mathbb{Z}\right\} \,.

The second 3\mathbb{Z}^3 \simeq \mathbb{Z}\oplus \mathbb{Z}\oplus \mathbb{Z} is the abelian group generated from the three (!) 1-simplicies in Δ[1]\Delta[1], namely the non-degenerate edge (01)(0\to 1) and the two degenerate cells (00)(0 \to 0) and (11)(1 \to 1), hence the abelian group of formal linear combinations of the form

3{a(00)+b(01)+c(11)|a,b,c}. \mathbb{Z}^3 \simeq \left\{ a \cdot (0\to 0) + b \cdot (0 \to 1) + c \cdot (1 \to 1) | a,b,c \in \mathbb{Z}\right\} \,.

The two face maps act on the basis 1-cells as

1:(ij)(i) \partial_1 \colon (i \to j) \mapsto (i)
0:(ij)(j). \partial_0 \colon (i \to j) \mapsto (j) \,.

Now of course most of the (infinitely!) many simplices inside Δ[1]\Delta[1] are degenerate. In fact the only non-degenerate simplices are the two 0-cells (0)(0) and (1)(1) and the 1-cell (01)(0 \to 1). Hence the alternating face maps complex modulo degeneracies, def. , of [Δ[1]]\mathbb{Z}[\Delta[1]] is simply this:

(C([Δ[1]]))/D([Δ[1]]))=(00(11) 2). (C (\mathbb{Z}[\Delta[1]])) / D (\mathbb{Z}[\Delta[1]])) = \left( \cdots \to 0 \to 0 \to \mathbb{Z} \stackrel{\left(1 \atop -1\right)}{\longrightarrow} \mathbb{Z}^2 \right) \,.

Notice that alternatively we could consider the topological 1-simplex Δ 1=[0,1]\Delta^1 = [0,1] and its singular simplicial complex Sing(Δ 1)Sing(\Delta^1) in place of the smaller Δ[1]\Delta[1], then the free simplicial abelian group (Sing(Δ 1))\mathbb{Z}(Sing(\Delta^1)) of that. The corresponding alternating face map chain complex C((Sing(Δ 1)))C(\mathbb{Z}(Sing(\Delta^1))) is “huge” in that in each positive degree it has a free abelian group on uncountably many generators. Quotienting out the degenerate cells still leaves uncountably many generators in each positive degree (while every singular nn-simplex in [0,1][0,1] is “thin”, only those whose parameterization is as induced by a degeneracy map are actually regarded as degenerate cells here). Hence even after normalization the singular simplicial chain complex is “huge”. Nevertheless it is quasi-isomorphic to the tiny chain complex found above.

The statement of the Dold-Kan correspondence now is the following.

Theorem

For AA an abelian category there is an equivalence of categories

N:A Δ opCh +(A):Γ N \;\colon\; A^{\Delta^{op}} \stackrel{\leftarrow}{\to} Ch_\bullet^+(A) \;\colon\; \Gamma

between

where

(Dold 58, Kan 58, Dold-Puppe 61).

Theorem (Kan)

For the case that AA is the category Ab of abelian groups, the functors NN and Γ\Gamma are nerve and realization with respect to the cosimplicial chain complex

[]:ΔCh +(Ab) \mathbb{Z}[-]: \Delta \to Ch_+(Ab)

that sends the standard nn-simplex to the normalized Moore complex of the free simplicial abelian group F (Δ n)F_{\mathbb{Z}}(\Delta^n) on the simplicial set Δ n\Delta^n, i.e.

Γ(V):[k]Hom Ch +(Ab)(N((Δ[k])),V). \Gamma(V) : [k] \mapsto Hom_{Ch_\bullet^+(Ab)}(N(\mathbb{Z}(\Delta[k])), V) \,.

This is due to (Kan 58).

More explicitly we have the following

Proposition
  • For VCh +V \in Ch_\bullet^+ the simplicial abelian group Γ(V)\Gamma(V) is in degree nn given by

    Γ(V) n= [n]surj[k]V k \Gamma(V)_n = \bigoplus_{[n] \underset{surj}{\to} [k]} V_k

    and for θ:[m][n]\theta : [m] \to [n] a morphism in Δ\Delta the corresponding map Γ(V) nΓ(V) m\Gamma(V)_n \to \Gamma(V)_m

    θ *: [n]surj[k]V k [m]surj[r]V r \theta^* : \bigoplus_{[n] \underset{surj}{\to} [k]} V_k \to \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    is given on the summand indexed by some σ:[n][k]\sigma : [n] \to [k] by the composite

    V kd *V s [m]surj[r]V r V_k \stackrel{d^*}{\to} V_s \hookrightarrow \bigoplus_{[m] \underset{surj}{\to} [r]} V_r

    where

    [m]t[s]d[k] [m] \stackrel{t}{\to} [s] \stackrel{d}{\to} [k]

    is the epi-mono factorization of the composite [m]θ[n]σ[k][m] \stackrel{\theta}{\to} [n] \stackrel{\sigma}{\to} [k].

  • The natural isomorphism ΓNId\Gamma N \to Id is given on AsAb Δ opA \in sAb^{\Delta^{op}} by the map

    [n]surj[k](NA) kA n \bigoplus_{[n] \underset{surj}{\to} [k]} (N A)_k \to A_n

    which on the direct summand indexed by σ:[n][k]\sigma : [n] \to [k] is the composite

    NA kA kσ *A n. N A_k \hookrightarrow A_k \stackrel{\sigma^*}{\to} A_n \,.
  • The natural isomorphism IdNΓId \to N \Gamma is on a chain complex VV given by the composite of the projection

    VC(Γ(V))C(Γ(C))/D(Γ(V)) V \to C(\Gamma(V)) \to C(\Gamma(C))/D(\Gamma(V))

    with the inverse

    C(Γ(V))/D(Γ(V))NΓ(V) C(\Gamma(V))/D(\Gamma(V)) \to N \Gamma(V)

    of

    NΓ(V)C(Γ(V))C(Γ(V))/D(Γ(V)) N \Gamma(V) \hookrightarrow C(\Gamma(V)) \to C(\Gamma(V))/D(\Gamma(V))

    (which is indeed an isomorphism, as discussed at Moore complex).

This is spelled out in (Goerss-Jardine, prop. 2.2 in section III.2).

Proposition

With the explicit choice for ΓNId\Gamma N \stackrel{\simeq}{\to} Id as above we have that Γ\Gamma and NN form an adjoint equivalence (ΓN)(\Gamma \dashv N)

This is for instance (Weibel, exercise 8.4.2).

Remark

It follows that with the inverse structure maps, we also have an adjunction the other way round: (NΓ)(N \dashv \Gamma).

Hence in conclusion the Dold-Kan correspondence allows us to regard chain complexes (in non-negative degree) as, in particular, special simplicial sets. In fact as simplicial sets they are Kan complexes and hence infinity-groupoids:

Theorem (J. C. Moore)

The simplicial set underlying any simplicial group (by forgetting the group structure) is a Kan complex.

This is due to (Moore, 1954)

In fact, not only are the horn fillers guaranteed to exist, but there is an algorithm that provides explicit fillers. This implies that constructions on a simplicial group that use fillers of horns can often be adjusted to be functorial by using the algorithmically defined fillers. An argument that just uses ‘existence’ of fillers can be refined to give something more ‘coherent’.

Proof

Let GG be a simplicial group.

Here is the explicit algorithm that computes the horn fillers:

Let (y 0,,y k1,,y k+1,,y n)(y_0,\ldots, y_{k-1}, -,y_{k+1}, \ldots, y_n) give a horn in G n1G_{n-1}, so the y iy_is are (n1)(n-1) simplices that fit together as if they were all but one, the k thk^{th} one, of the faces of an nn-simplex. There are three cases:

  1. if k=0k = 0:

    • Let w n=s n1y nw_n = s_{n-1}y_n and then w i=w i+1(s i1d iw i+1) 1s i1y iw_i = w_{i+1}(s_{i-1}d_i w_{i+1})^{-1}s_{i-1}y_i for i=n,,1i = n, \ldots, 1, then w 1w_1 satisfies d iw 1=y id_i w_1 = y_i, i0i\neq 0;
  2. if 0<k<n0\lt k \lt n:

    • Let w 0=s 0y 0w_0 = s_0 y_0 and w i=w i1(s id iw i1) 1s iy iw_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i for i=0,,k1i = 0, \ldots, k-1, then take w n=w k1(s n1d nwk1) 1s n1y nw_n = w_{k-1}(s_{n-1}d_nw_{k-1})^{-1}s_{n-1}y_n, and finally a downwards induction given by w i=w i+1(s i1d iw i+1) 1s i1y iw_i = w_{i+1}(s_{i-1}d_{i}w_{i+1})^{-1}s_{i-1}y_i, for i=n,,k+1i = n, \ldots, k+1, then w k+1w_{k+1} gives d iw k+1=y id_{i}w_{k+1} = y_i for iki \neq k;
  3. if k=nk=n:

    • use w 0=s 0y 0w_0 = s_0 y_0 and w i=w i1(s id iw i1) 1s iy iw_i = w_{i-1}(s_i d_i w_{i-1})^{-1}s_i y_i for i=0,,n1i = 0, \ldots, n-1, then w n1w_{n-1} satisfies d iw n1=y id_i w_{n-1} = y_i, ini\neq n.

Geometric realization

So far we we have considered passing from topological spaces to simplicial sets by applying the singular simplicial complex functor of def. . Now we discuss a left adjoint of this functor, called geometric realization, which turns a simplicial set into a topological space by identifying each of its abstract n-simplices with the standard topological nn-simplex.

This is an example of a general abstract phenomenon:

Proposition

Let

δ:D𝒞 \delta \;\colon\; D \longrightarrow \mathcal{C}

be a functor from a small category DD to a locally small category 𝒞\mathcal{C} with all colimits. Then the nerve-functor

N:𝒞[D op,Set] N \;\colon\; \mathcal{C} \longrightarrow [D^{op}, Set]
N(X)𝒞(δ(),X) N(X) \coloneqq \mathcal{C}(\delta(-),X)

has a left adjoint functor ||{\vert-\vert}, called geometric realization,

(||N):𝒞N||[D op,Set] ({\vert-\vert} \dashv N) \;\colon\; \mathcal{C} \stackrel{\overset{{\vert-\vert}}{\longleftarrow}}{\underset{N}{\longrightarrow}} [D^{op}, Set]

given by the coend

|S|= dDδ(d)S(d). {\vert S\vert} = \int^{d \in D} \delta(d) \cdot S(d) \,.

(Kan 58)

Proof

By basic propeties of ends and coends:

[D op,Set](S,N(X)) = dDSet(S(d),N(X)(d)) = dDSet(S(d),𝒞(δ(d),X)) dD𝒞(δ(d)S(d),X) 𝒞( dDδ(d)S(d),X) =𝒞(|S|,X). \begin{aligned} [D^{op}, Set](S,N(X)) & = \int_{d \in D} Set(S(d), N(X)(d)) \\ & = \int_{d\in D} Set(S(d), \mathcal{C}(\delta(d),X)) \\ & \simeq \int_{d \in D} \mathcal{C}(\delta(d) \cdot S(d), X) \\ & \simeq \mathcal{C}(\int^{d \in D} \delta(d) \cdot S(d), X) \\ & = \mathcal{C}({\vert S\vert}, X) \,. \end{aligned}
Example

The singular simplicial complex functor SingSing of def. has a left adjoint geometric realization functor

||:sSetTop {\vert-\vert} \colon sSet \longrightarrow Top

given by the coend

|S|= [n]ΔΔ nS n. {\vert S \vert} = \int^{[n]\in \Delta} \Delta^n \cdot S_n \,.

Topological geometric realization takes values in particularly nice topological spaces.

Proposition

The topological geometric realization of simplicial sets in example takes values in CW-complexes.

(e.g. Goerss-Jardine 99, chapter I, prop. 2.3)

Thus for a topological space XX the adjunction counit ϵ X:|SingX|X\epsilon_X \colon {\vert Sing X\vert} \longrightarrow X of the nerve and realization-adjunction is a candidate for a replacement of XX by a CW-complex. For this, ϵ X\epsilon_X should be at least a weak homotopy equivalence, i.e. induce isomorphisms on all homotopy groups. Since homotopy groups are built from maps into XX out of compact topological spaces it is plausible that this works if the topology of XX is entirely detected by maps out of compact topological spaces into XX. Topological spaces with this property are called compactly generated.

We take compact topological space to imply Hausdorff topological space.

Definition

A subspace UXU \subset X of a topological space XX is called compactly open or compactly closed, respectively, if for every continuous function f:KXf \colon K \longrightarrow X out of a compact topological space the preimage f 1(U)Kf^{-1}(U) \subset K is open or closed, respectively.

A topological space XX is a compactly generated topological space if each of its compactly closed subspaces is already closed.

Write

Top cgTop Top_{cg} \hookrightarrow Top

for the full subcategory of Top on the compactly generated topological spaces.

Often the condition is added that a compactly closed topological space be also a weakly Hausdorff topological space.

Corollary

The topological geometric realization functor of simplicial sets in example takes values in compactly generated topological spaces

||:sSetTop cg {\vert - \vert} \;\colon\; sSet \longrightarrow Top_{cg}
Proof

By example and prop. .

Proposition

The subcategory Top cgTopTop_{cg} \hookrightarrow Top of def. has the following properties

  1. It is a coreflective subcategory

    Top cgkTop. Top_{cg} \stackrel{\hookrightarrow}{\underset{k}{\longleftarrow}} Top \,.

    The coreflection k(X)k(X) of a topological space is given by adding to the open subsets of XX all compactly open subsets, def. .

  2. It has all small limits and colimits.

    The colimits are computed in TopTop, the limits are the image under kk of the limits as computed in TopTop.

  3. It is a cartesian closed category.

    The cartesian product in Top cgTop_{cg} is the image under kk of the Cartesian product formed in TopTop.

This is due to (Steenrod 67), expanded on in (Lewis 78, appendix A). One says that prop. with example makes Top cgTop_{cg} a “convenient category of topological spaces”.

Proposition

Regarded, via corollary as a functor ||:sSetTop cg{\vert - \vert} \colon sSet \to Top_{cg}, geometric realization preserves finite limits.

See at Geometric realization is left exact.

Proof idea

The key step in the proof is to use the cartesian closure of Top cgTop_{cg} (prop. ). This gives that the Cartesian product is a left adjoint and hence preserves colimits in each variable, so that the coend in the definition of the geometric realization may be taken out of Cartesian products.

Lemma

The geometric realization, example , of a minimal Kan fibration, def. is a Serre fibration, def. .

This is due to (Gabriel-Zisman 67). See for instance (Goerss-Jardine 99, chapter I, corollary 10.8, theorem 10.9).

Proof idea

By prop. minimal Kan fibrations are simplicial fiber bundles, locally trivial over each simplex in the base. By prop. this property translates to their geometric realization also being a locally trivial fiber bundle of topological spaces, hence in particular a Serre fibration.

Proposition

The geometric realization, example , of any Kan fibration, def. is a Serre fibration, def. .

This is due to (Quillen 68). See for instance (Goerss-Jardine 99, chapter I, theorem 10.10).

Proposition

For SS a Kan complex, then the unit of the nerve and realization-adjunction (prop. , example )

SSing|S| S \longrightarrow Sing {\vert S \vert}

is a weak homotopy equivalence, def. .

For XX any topological space, then the adjunction counit

|SingX|X {\vert Sing X\vert} \longrightarrow X

is a weak homotopy equivalence

e.g. (Goerss-Jardine 99, chapter I, prop. 11.1 and p. 63).

Proof idea

Use prop. and prop. applied to the path fibration to proceed by induction.

\,

The classical model structure on simplicial sets

Definition

(classical model structure on simplicial sets)

The classical model structure on simplicial sets, sSet QuillensSet_{Quillen}, has the following distinguished classes of morphisms:

Proposition

In model structure sSet QuillensSet_{Quillen}, def. , the following holds.

Proposition

The acyclic fibrations in sSet QuillensSet_{Quillen}(i.e. the maps that are both fibrations as well as weak equivalences) between Kan complexes are precisely the morphisms f:XYf : X \to Y that have the right lifting property with respect to all inclusions Δ[n]Δ[n]\partial \Delta[n] \hookrightarrow \Delta[n] of boundaries of nn-simplices into their nn-simplices

Δ[n] X f Δ[n] Y. \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.

This appears spelled out for instance as (Goerss-Jardine 99, theorem 11.2).

In fact:

Proposition

sSet QuillensSet_{Quillen} is a cofibrantly generated model category with

  • generating cofibrations the boundary inclusions Δ[n]Δ[n]\partial \Delta[n] \to \Delta[n];

  • generating acyclic cofibrations the horn inclusions Λ i[n]Δ[n]\Lambda^i[n] \to \Delta[n].

Theorem

Let WW be the smallest class of morphisms in sSetsSet satisfying the following conditions:

  1. The class of monomorphisms that are in WW is closed under pushout, transfinite composition, and retracts.
  2. WW has the two-out-of-three property in sSetsSet and contains all the isomorphisms.
  3. For all natural numbers nn, the unique morphism Δ[n]Δ[0]\Delta [n] \to \Delta [0] is in WW.

Then WW is the class of weak homotopy equivalences.

Proof
  • First, notice that the horn inclusions Λ 0[1]Δ[1]\Lambda^0 [1] \hookrightarrow \Delta [1] and Λ 1[1]Δ[1]\Lambda^1 [1] \hookrightarrow \Delta [1] are in WW.
  • Suppose that the horn inclusion Λ k[m]Δ[m]\Lambda^k [m] \hookrightarrow \Delta [m] is in WW for all m<nm \lt n and all 0km0 \le k \le m. Then for 0ln0 \le l \le n, the horn inclusion Λ l[n]Δ[n]\Lambda^l [n] \hookrightarrow \Delta [n] is also in WW.
  • Quillen’s small object argument then implies all the trivial cofibrations are in WW.
  • If p:XYp : X \to Y is a trivial Kan fibration, then its right lifting property implies there is a morphism s:YXs : Y \to X such that ps=id Yp \circ s = id_Y, and the two-out-of-three property implies s:YXs : Y \to X is a trivial cofibration. Thus every trivial Kan fibration is also in WW.
  • Every weak homotopy equivalence factors as pip \circ i where pp is a trivial Kan fibration and ii is a trivial cofibration, so every weak homotopy equivalence is indeed in WW.
  • Finally, noting that the class of weak homotopy equivalences satisfies the conditions in the theorem, we deduce that it is the smallest such class.

As a corollary, we deduce that the classical model structure on sSetsSet is the smallest (in terms of weak equivalences) model structure for which the cofibrations are the monomorphisms and the weak equivalences include the (combinatorial) homotopy equivalences.

Proposition

Let π 0:sSetSet\pi_0 : sSet \to Set be the connected components functor, i.e. the left adjoint of the constant functor cst:SetsSetcst : Set \to sSet. A morphism f:ZWf : Z \to W in sSetsSet is a weak homotopy equivalence if and only if the induced map

π 0K f:π 0K Wπ 0K Z\pi_0 K^f : \pi_0 K^W \to \pi_0 K^Z

is a bijection for all Kan complexes KK.

Proof

One direction is easy: if KK is a Kan complex, then axiomS FOR simplicial model categories (Def. ) implies the functor K ():sSet opsSetK^{(-)} : sSet^{op} \to sSet is a right Quillen functor, so Ken Brown's lemma (Prop. ) implies that it preserves all weak homotopy equivalences; in particular, π 0K ():sSet opSet\pi_0 K^{(-)} : sSet^{op} \to Set sends weak homotopy equivalences to bijections.

Conversely, when KK is a Kan complex, there is a natural bijection between π 0K X\pi_0 K^X and the hom-set Ho(sSet)(X,K)Ho (sSet) (X, K), and thus by the Yoneda lemma, a morphism f:ZWf : Z \to W such that the induced morphism π 0K Wπ 0K Z\pi_0 K^W \to \pi_0 K^Z is a bijection for all Kan complexes KK is precisely a morphism that becomes an isomorphism in Ho(sSet)Ho (sSet), i.e. a weak homotopy equivalence.

Theorem

(Quillen equivalence between classical model structure on topological spaces and classical model structure on simplicial sets)

The singular simplicial complex/geometric realization-adjunction of example constitutes a Quillen equivalence between the classical model structure on simplicial sets sSet QuillensSet_{Quillen} of def. and the classical model structure on topological spaces:

(||Sing):Top Quillen Q QSing||sSet Quillen ({\vert -\vert}\dashv Sing) \;\colon\; Top_{Quillen} \underoverset {\underset{Sing}{\longrightarrow}} {\overset{{\vert -\vert}}{\longleftarrow}} {\phantom{{}_{Q}}\simeq_{Q}} sSet_{Quillen}
Proof

First of all, the adjunction is indeed a Quillen adjunction: prop. says in particular that Sing()Sing(-) takes Serre fibrations to Kan fibrations and prop. gives that ||{\vert-\vert} sends monomorphisms of simplicial sets to relative cell complexes.

Now prop. says that the derived adjunction unit and derived adjunction counit are weak equivalences, and hence the Quillen adjunction is a Quillen equivalence.

\,

Basic notions of higher topos theory

We have discussed basic notions of topos theory above and of homotopy theory (above). The combination of the two is higher topos theory which we discuss here.

We had explained how toposes may be thought of as categories of generalized spaces and how homotopy theory is about relaxing the concept of equality to that of gauge transformation/homotopy and higher gauge transformation/higher homotopy. Accordingly, higher toposes may be thought of as higher categories of generalized spaces whose probe are defined only up to gauge transformation/homotopy. Examples of such include orbifolds and Lie groupoids.

(…)

Locally presentable \infty-Categories

The analog of the notion of locally presentable categories (Def. ) for model categories (Def. ) are combinatorial model categories (Def. ) below. In addition to the ordinary condition of presentability of the underlying category, these are required to be cofibrant generation (Def. below) in that all cofibrations are retracts of relative cell complex-inclusions.

That this is indeed the correct model category-analog of locally presentable categories is the statement of Dugger's theorem (Def. below).

Hence as we pass to the localization of the very large category of combinatorial model categories at the Quillen equivalences, we obtain a homotopy-theoretic refinement of the very large category PrCat of locally presentable categories: Ho(CombModCat) (Def. ). An object in Ho(CombModCat) we also refer to as a locally presentable (∞,1)-category, and a morphism in Ho(CombModCat) we also refer to as the equivalence class of an (∞,1)-colimit-preserving (∞,1)-functor.

\,

Definition

(cofibrantly generated model category)

A model category 𝒞\mathcal{C} (def. ) is called cofibrantly generated if there exists two small subsets

I,JMor(𝒞) I, J \subset Mor(\mathcal{C})

of its class of morphisms, such that

  1. II and JJ have small domains according to def. ,

  2. the (acyclic) cofibrations of 𝒞\mathcal{C} are precisely the retracts, of II-relative cell complexes (JJ-relative cell complexes), def. .

Proposition

For 𝒞\mathcal{C} a cofibrantly generated model category, def. , with generating (acylic) cofibrations II (JJ), then its classes W,Fib,CofW, Fib, Cof of weak equivalences, fibrations and cofibrations are equivalently expressed as injective or projective morphisms (def. ) this way:

  1. Cof=(IInj)ProjCof = (I Inj) Proj

  2. WFib=IInjW \cap Fib = I Inj;

  3. WCof=(JInj)ProjW \cap Cof = (J Inj) Proj;

  4. Fib=JInjFib = J Inj;

Proof

It is clear from the definition that I(IInj)ProjI \subset (I Inj) Proj, so that the closure property of prop. gives an inclusion

Cof(IInj)Proj. Cof \subset (I Inj) Proj \,.

For the converse inclusion, let f(IInj)Projf \in (I Inj) Proj. By the small object argument, prop. , there is a factorization f:ICellIInjf\colon \overset{\in I Cell}{\longrightarrow}\overset{I Inj}{\longrightarrow}. Hence by assumption and by the retract argument lemma , ff is a retract of an II-relative cell complex, hence is in CofCof.

This proves the first statement. Together with the closure properties of prop. , this implies the second claim.

The proof of the third and fourth item is directly analogous, just with JJ replaced for II.

Example

(category of simplicial presheaves)

Let 𝒞\mathcal{C} be a small (Def. ) sSet-enriched category (Def. with Example ) and consider the enriched presheaf category (Example )

sPSh(𝒞)[𝒞 op,sSet] sPSh(\mathcal{C}) \;\coloneqq\; [\mathcal{C}^{op}, sSet]

This is called the category of simplicial presheaves on 𝒞\mathcal{C}.

By Prop. this is equivalent (Def. ) to the category of simplicial objects in the category of presheaves over 𝒞\mathcal{C} (Example ):

(96)[𝒞 op,sSet][Δ op,𝒞 op,Set] [\mathcal{C}^{op}, sSet] \;\simeq\; [\Delta^{op}, \mathcal{C}^{op}, Set]

This implies for instance that if

𝒟F𝒟 \mathcal{D} \overset{F}{\longrightarrow} \mathcal{D}

a functor, the induced adjoint triple (Remark ) of sSet-enriched functor Kan extensions (Prop. )

[𝒞 op,sSet]AAAALan F AAAAF * AAAARan F[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \; \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{Lan_F}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{F^\ast}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{Ran_F}{\longrightarrow} } \; [\mathcal{D}^{op}, sSet]

is given simplicial-degreewise by the corresponding Set-enriched Kan extensions.

Proposition

(model categories of simplicial presheaves)

Let 𝒞\mathcal{C} be a small (Def. ) sSet-enriched category (Def. with Example ). Then the category of simplicial presheaves [𝒞 op,sSet][\mathcal{C}^{op}, sSet] (Example ) carries the following two structures of a model category (Def. )

  1. the projective model structure on simplicial presheaves

    [𝒞 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj}

    has as weak equivalences and fibrations those natural transformations η\eta whose component on every object c𝒞c \in \mathcal{C} is a weak equivalences or fibration, respectively, in the classical model structure on simplicial sets (Def. );

  2. the injective model structure on simplicial presheaves

    [𝒞 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj}

    has as weak equivalences and cofibrations those natural transformations η\eta whose component on every object c𝒞c \in \mathcal{C} is a weak equivalences or cofibration, respectively, in the classical model structure on simplicial sets (Def. );

Moreover, the identity functors constitute a Quillen equivalence (Def. ) between these two model structures

(97)[𝒞 op,sSet Qu] inj QuAAidAAAAidAA[𝒞 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}
Remark

The Quillen adjunction (97) in Prop. implies in particular that

  1. every projective cofibration is in particular an objectwise cofibration;

  2. every injective fibration is in particular an objectwise fibration;

Proposition

(some projectively cofibrant simplicial presheaves)

Let 𝒞\mathcal{C} be a small (Def. ). Then a sufficient condition for a simplicial presheaf over 𝒞\mathcal{C} (Def. )

X[𝒞 op,sSet Qu] proj \mathbf{X} \;\in\; [\mathcal{C}^{op}, sSet_{Qu}]_{proj}

to be a cofibrant object with respect to the projective model structure on simplicial presheaves (Prop. ) is that

  1. X\mathbf{X} is degreewise a coproduct of representable presheaves

    X ki ky(X i k) \mathbf{X}_k \;\simeq\; \underset{i_k}{\coprod} y(X_{i_k})
  2. the degeneracy maps are inclusions of direct summands.

In particular every representable presheaf, regarded as a simplicially constant simplicial presheaf, is projectively cofibrant.

(Dugger 00, section 9, lemma 2.7)

The following concept of left Bousfield localization is the analog for model categories of the concept of reflection onto local objects (Def. ):

Definition

(left Bousfield localization)

A left Bousfield localization 𝒞 loc\mathcal{C}_{loc} of a model category 𝒞\mathcal{C} (Def. ) is another model category structure on the same underlying category with the same cofibrations,

Cof loc=Cof Cof_{loc} = Cof

but more weak equivalences

W locW. W_{loc} \supset W \,.

We say that this is localization at W locW_{loc}.

Notice that:

Proposition

(left Bousfield localization is Quillen reflection)

Given a left Bousfield localization 𝒞 loc\mathcal{C}_{loc} of 𝒞\mathcal{C} as in def. , then the identity functor exhibits a Quillen reflection (Def. )

𝒞 loc Qu QuAAidAAid𝒞. \mathcal{C}_{loc} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C} \,.

In particular, by Prop. , the induced adjunction of derived functors (Prop. ) exhibits a reflective subcategory inclusion of homotopy categories (Def. )

Ho(𝒞 loc)AAidAA𝕃idHo(𝒞). Ho(\mathcal{C}_{loc}) \underoverset {\underset{\phantom{AA}\mathbb{R} id \phantom{AA}}{\hookrightarrow}} {\overset{\mathbb{L}id}{\longleftarrow}} {\bot} Ho(\mathcal{C}) \,.
Proof

We claim that

  1. Fib locFibFib_{loc} \subset Fib;

  2. W locFib loc=WFibW_{loc} \cap Fib_{loc} = W \cap Fib;

Using the properties of the weak factorization systems (def.) of (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations) for both model structures we get

Fib loc =(Cof locW loc)Inj (Cof locW)Inj =Fib \begin{aligned} Fib_{loc} &= (Cof_{loc} \cap W_{loc})Inj \\ &\subset (Cof_{loc} \cap W)Inj \\ & = Fib \end{aligned}

and

Fib locW loc =Cof locInj =CofInj =FibW. \begin{aligned} Fib_{loc} \cap W_{loc} & = Cof_{loc} Inj \\ & = Cof \, Inj \\ & = Fib \cap W \end{aligned} \,.

Next to see that the identity functor constitutes a Quillen adjunction (Def. ): By construction, id:𝒞𝒞 locid \colon \mathcal{C} \to \mathcal{C}_{loc} preserves cofibrations and acyclic cofibrations, hence is a left Quillen functor.

To see that the derived adjunction counit (Def. ) is a weak equivalence:

Since we have an adjoint pair of identity functors, the ordinary adjunction counit is the identity morphisms and hence the derived adjunction counit on a fibrant object cc is just a cofibrant resolution-morphism

Q(c)W 𝒟Fib 𝒟p cc Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c

but regarded in the model structure 𝒟 loc\mathcal{D}_{loc}. Hence it is sufficient to see that acyclic fibrations in 𝒟\mathcal{D} remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, bu the first point above.

We may also easily check directly the equivalent statement (via Prop. ) that the induced adjunction of derived functors on homotopy categories is a reflective subcategory-inclusion:

Since Cof loc=CofCof_{loc} = Cof the notion of left homotopy in 𝒞 loc\mathcal{C}_{loc} is the same as that in 𝒞\mathcal{C}, and hence the inclusion of the subcategory of local cofibrant-fibrant objects into the homotopy category of the original cofibrant-fibrant objects is clearly a full subcategory inclusion. Since Fib locFibFib_{loc} \subset Fib by the first statement above, on these cofibrant-fibrant objects the right derived functor of the identity is just the identity and hence does exhibit this inclusion. The left adjoint to this inclusion is given by 𝕃id\mathbb{L}id, by the general properties of Quillen adjunctions (Prop. )).

Proof

We consider the case of left Bousfield localizations, the other case is formally dual.

A left Bousfield localization is a Quillen adjunction by identity functors (this Remark)

𝒟 loc Qu QuAAidAAid𝒟 \mathcal{D}_{loc} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {{}_{\phantom{Qu}} \bot_{Qu}} \mathcal{D}

This means that the ordinary adjunction counit is the identity morphisms and hence that the derived adjunction counit on a fibrant object cc is just a cofibrant resolution-morphism

Q(c)W 𝒟Fib 𝒟p cc Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c

but regarded in the model structure 𝒟 loc\mathcal{D}_{loc}. Hence it is sufficient to see that acyclic fibrations in 𝒟\mathcal{D} remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, by this Remark.

The following proposition says that Definition of combinatorial model categories is indeed the suitable analog of Def. of locally presentable categories:

Proposition

(Dugger's theorem)

Let 𝒞\mathcal{C} be a combinatorial model category (Def. ). Then there exists

  1. a small category 𝒮\mathcal{S};

  2. a small set SMor [𝒮 op,sSet]S \subset Mor_{[\mathcal{S}^{op}, sSet]} in its category of simplicial presheaves (Example );

1 a Quillen equivalence (Def. )

[𝒮 op,sSet Qu] proj,S Qu QuAAAA𝒞 [\mathcal{S}^{op}, sSet_{Qu}]_{proj,S} \underoverset {\underset{\phantom{AAAA}}{\longrightarrow}} {\overset{}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}

between 𝒞\mathcal{C} and the left Bousfield localization (Def. ) of the projective model structure on simplicial presheaves over 𝒞\mathcal{C} (Prop. ) at the set SS.

Definition

(homotopy category of presentable (∞,1)-categories)

Write CombModCatCombModCat for the very large category whose objects are combinatorial model categories (Def. ) and whose morphisms are left Quillen functors between them (Def. ).

We write

Ho(CombModCat)CombModCat[QuillenEquivs 1]\coloneqq CombModCat\big[ QuillenEquivs^{-1}\big]

for its localization (Def. ) at the Quillen equivalences (Def. ).

We say:

The following example is the genralization of the category of sets (Def. ) as we pass to homotopy theory:

Example

(∞Grpd)

The image of the classical model structure on simplicial sets sSet QusSet_{Qu} (Def. ), which is combinatorial model category by example , under the localization to Ho(CombModCat) (Def. ), we call the presentable (∞,1)-category of ∞-groupoids:

CombModCat AAγAA Ho(CombModCat) sSet Qu Grpd \array{ CombModCat &\overset{\phantom{AA}\gamma\phantom{AA}}{\longrightarrow}& Ho(CombModCat) \\ sSet_{Qu} &\mapsto& \infty Grpd }

In order to get good control over left Bousfield localization (Def. ) and hence over presentable (∞,1)-categories (Def. ) we need the analog of Prop. , saying that reflective localization are reflections onto their full subcategories of local objects. For this, in turn, we need a good handle on the hom-infinity-groupoids:

Definition

(simplicial model category)

An sSet Quillen{}_{Quillen}-enriched model category or simplicial model category, for short is a category 𝒞\mathcal{C} (Def. ) equipped with

  1. the structure of an sSet-enriched category (Def. via Example ), which is also tensored and cotensored over sSet (Def. )

    (with sSet (Def. ), equipped with its canonical structure of a cosmos from Prop. , Example ),

  2. the structure of a model category (Def. )

such that these two structures are compatible in the following way:

Proposition

(in simplicial model category enriched hom-functor out of cofibrant into fibrant is homotopical functor)

Let 𝒞\mathcal{C} be a simplicial model category (Def. ).

If Y𝒞Y \in \mathcal{C} is a cofibrant object, then the enriched hom-functor (Example ) out of XX

𝒞(Y,):𝒞sSet Qu \mathcal{C}(Y,-) \;\colon\; \mathcal{C} \longrightarrow sSet_{Qu}

preserves fibrations and acyclic fibrations.

If A𝒞A \in \mathcal{C} is a fibrant object, then the enriched hom-functor (Example ) into XX

𝒞(,A):𝒞 opsSet Qu \mathcal{C}(-,A) \;\colon\; \mathcal{C}^{op} \longrightarrow sSet_{Qu}

sends cofibrations and acyclic cofibrations in 𝒞\mathcal{C} to fibrations and acyclic fibrations, respectively, in the classical model structure on simplicial sets.

Proof

In the first case, consider the comparison morphism (98) for X=X =\emptyset the initial object, in the second case consider it for B=*B = \ast the terminal object (Def. )

Since 𝒞\mathcal{C} is a tensored and cotensored category, Prop. says that

𝒞(,)*AAandAA𝒞(,*)*sSet. \mathcal{C}(\emptyset, -) \;\simeq\; \ast \phantom{AA} \text{and} \phantom{AA} \mathcal{C}(-,\ast) \;\ast\; \;\;\; \in sSet \,.

This means that in the first case the comparison morphism

𝒞(Y,A)𝒞(X,A)×𝒞(X,B)𝒞(Y,B) \mathcal{C}(Y,A) \longrightarrow \mathcal{C}(X,A) \underset{\mathcal{C}(X,B)}{\times} \mathcal{C}(Y,B)

(98) becomes equal to the top morphism in the following diagram

𝒞(Y,A) 𝒞(Y,g) 𝒞(Y,B) * AAA * \array{ \mathcal{C}(Y,A) &\overset{\mathcal{C}(Y,g)}{\longrightarrow}& \mathcal{C}(Y,B) \\ \Big\downarrow && \Big\downarrow \\ \ast &\underset{\phantom{AAA}}{\longrightarrow}& \ast }

while in the second case it becomes equal to the left morphism in

𝒞(Y,A) 𝒞(Y,g) * 𝒞(f,A) 𝒞(X,A) AAA * \array{ \mathcal{C}(Y,A) &\overset{\phantom{\mathcal{C}(Y,g)}}{\longrightarrow}& \ast \\ {}^{\mathllap{ \mathcal{C}(f,A) }}\Big\downarrow && \Big\downarrow \\ \mathcal{C}(X,A) &\underset{\phantom{AAA}}{\longrightarrow}& \ast }

Hence the claim follows by the defining condition on the comparison morphism in a simplicial model category.

Definition

(derived hom-functor)

Let 𝒞\mathcal{C} be a simplicial model category (Def. ).

By Prop. and by Ken Brown's lemma (Prop. ), the enriched hom-functor (Example ) has a right derived functor (Def. ) when its first argument is cofibrant and its second argument is fibrant. The combination is called the derived hom-functor

hom:Ho(𝒞) op×Ho(𝒞)Ho(sSet Quillen) \mathbb{R}hom \;\colon\; Ho(\mathcal{C})^{op} \times Ho(\mathcal{C}) \longrightarrow Ho(sSet_{Quillen})

In view of the Quillen equivalence sSet Qu QuTop QusSet_{Qu} \simeq_{Qu} Top_{Qu} (Theorem ), the simplicial sets (Kan complexes) hom(X,A)\mathbb{R}hom(X,A) are also called the derived hom-spaces.

In the presence of functorial cofibrant resolution QQ and fibrant resolution PP (Def. ) this is given by the ordinary enriched hom-functor 𝒞(,)\mathcal{C}(-,-) as

hom(X,Y)𝒞(QX,PY). \mathbb{R}hom(X,Y) \;\simeq\; \mathcal{C}(Q X, P Y) \,.
Proposition

(recognition of simplicial Quillen adjunctions)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be two simplicial model categories (Def. ) such that 𝒟\mathcal{D} is also a left proper model category (Def. ). Then for an sSet-enriched adjunction (Def. ) of the form

𝒞RAALAA𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{D}

to be Quillen adjunction (Def. , hence a simplicial Quillen adjunction) it is sufficient that the following two conditions hold:

  1. LL preserves cofibrations,

  2. RR preserves fibrant objects

(i.e. this already implies that RR preserves all fibrations).

(Lurie HTT, cor. A.3.7.2)

Proposition

(model structure on simplicial presheaves is left proper combinatorial simplicial model category)

Let 𝒞\mathcal{C} be a small (Def. ) sSet-enriched category (Def. with Example ). Then the injective and projective model structure on simplicial presheaves over 𝒞\mathcal{C} (Prop. )

[𝒞 op,sSet Qu] proj,A[𝒞 op,sSet Qu] injCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \;, \phantom{A} [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \;\;\; \in CombModCat

are

  1. proper model categories (Def. ),

  2. simplicial model categories (Def. ),

  3. combinatorial model categories (Def. ).

The following is the model category-analog of the concept of local objects from Def. :

Definition

(local objects and local morphisms in a model category)

Let 𝒞\mathcal{C} be a simplicial model category (Def. ) and let SMor 𝒞S \subset Mor_{\mathcal{C}} be a sub-class of its class of morphisms. Then

  1. an object A𝒞A \in \mathcal{C} is called a (derived-)local object if for every XsYSX \overset{s}{\to} Y \; \in S the value of the derived hom-functor (Def. ) out of ss into XX is a weak equivalence (i.e. an isomorphism in the classical homotopy category Ho(sSet)Ho(sSet))

    Hom(s,A):Hom(Y,A)Hom(X,A) \mathbb{R}Hom(s,A) \;\colon\; \mathbb{R}Hom(Y,A) \overset{\simeq}{\longrightarrow} \mathbb{R}Hom(X,A)
  2. a morphism XfYX \overset{f}{\to} Y in 𝒞\mathcal{C} is called a (derived-)local morphism if for every local object AA we have

    Hom(f,A):Hom(Y,A)Hom(X,A) \mathbb{R}Hom(f,A) \;\colon\; \mathbb{R}Hom(Y,A) \overset{\simeq}{\longrightarrow} \mathbb{R}Hom(X,A)

The following is the model category-analog of the characterization from Prop. of reflective localizations as reflections onto local objects:

Proposition

(existence of left Bousfield localization for left proper simplicial combinatorial model categories)

Let 𝒞\mathcal{C} be a combinatorial model category (Def. ) which is left proper (Def. ) and simplicial (Def. ), and let SMor 𝒞S \subset Mor_{\mathcal{C}} be a small set of its morphisms.

Then the left Bousfield localization (Def. ) of 𝒞\mathcal{C} at SS, namely at the class of SS-local morphisms (Def. ) exist, to be denoted L S𝒞L_S \mathcal{C}, and it has the following properties:

  1. L S𝒞L_S \mathcal{C} is itself a left proper simplicial combinatorial model category;

  2. the fibrant objects of L S𝒞L_S \mathcal{C} are precisely those fibrant objects of 𝒞\mathcal{C} which in addition are SS-local objects (Def. );

  3. the homotopy category (Def. ) of L S𝒞L_S \mathcal{C} is the full subcategory of that of 𝒞\mathcal{C} on ( the images under localization of) the SS-local objects.

    Ho(L S𝒞)Ho(𝒞) Ho(L_S \mathcal{C}) \hookrightarrow Ho(\mathcal{C})

The following class of examples of left Bousfield localizations generalizes those of Def. from 1-categories to locally presentable (∞,1)-categories:

Definition

(homotopy localization of combinatorial model categories)

Let 𝒞\mathcal{C} be a combinatorial model category (Def. ) which, by Dugger's theorem (Prop. ) is Quillen equivalent to a left Bousfield localization of a model category of simplicial presheaves over some small simplicial category 𝒮\mathcal{S}

𝒞 QuAAidAAAAidAA[𝒮 op,sSet Qu] projCombModCati.e.𝒞AALAAPSh (𝒮)Ho(CombModCat) \mathcal{C} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} [\mathcal{S}^{op}, sSet_{Qu}]_{proj} \; \in CombModCat \;\text{i.e.}\; \mathcal{C} \underoverset {\underset{}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{S}) \; \in Ho(CombModCat)

Let moreover

𝔸[𝒮 op,sSet Qu] \mathbb{A} \in [\mathcal{S}^{op}, sSet_{Qu}]

be any object. Then the homotopy localization of 𝒞\mathcal{C} at 𝔸\mathbb{A} is the further left Bousfield localization (Def. ) at the morphisms of the form

X×𝔸p 1X X \times \mathbb{A} \overset{p_1}{\longrightarrow} X

for all X𝒮X \in \mathcal{S}:

[𝒮 op,sSet Qu] proj,𝔸 QuAAidAAAAidAA[𝒮 op,sSet Qu] proj QuAAidAAAAidAA𝒞CombModCat. [\mathcal{S}^{op}, sSet_{Qu}]_{proj, \mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} [\mathcal{S}^{op}, sSet_{Qu}]_{proj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\simeq_{Qu}} \mathcal{C} \;\;\;\; \in CombModCat \,.

The image of this homotopy localization in Ho(CombModCat) (Def. ) we denote by

𝒞 𝔸ιL 𝔸𝒞Ho(CombModCat). \mathcal{C}_{\mathbb{A}} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} \mathcal{C} \;\;\; \in Ho(CombModCat) \,.

\infty-Modalities

The following is an homotopy theoretic analog of adjoint triples (Remark ):

Definition

(Quillen adjoint triple)

Let 𝒞 1,𝒞 2,𝒟\mathcal{C}_1, \mathcal{C}_2, \mathcal{D} be model categories (Def. ), where 𝒞 1\mathcal{C}_1 and 𝒞 2\mathcal{C}_2 share the same underlying category 𝒞\mathcal{C}, and such that the identity functor on 𝒞\mathcal{C} constitutes a Quillen equivalence (Def. )

(99)𝒞 2 Qu QuAAidAAAAidAA𝒞 1 \mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{C}_1

Then a Quillen adjoint triple

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

is a pair of Quillen adjunctions (Def. ), as shown, together with a 2-morphism in the double category of model categories (Def. )

(100)𝒟 ACA 𝒞 1 C id id 𝒞 2 AAidAA 𝒞 2 \array{ \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }

whose derived natural transformation Ho(id)Ho(id) (Def. ) is invertible (a natural isomorphism).

If two Quillen adjoint triples overlap

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

we speak of a Quillen adjoint quadruple, and so forth.

Proposition

(Quillen adjoint triple induces adjoint triple of derived functors on homotopy categories)

Given a Quillen adjoint triple (Def. ), the induced derived functors (Def. ) on the homotopy categories (Def. ) form an ordinary adjoint triple (Remark ):

𝒞 1/2 Qu QuL Qu QuC AARAA 𝒟AAAAHo()AAAAHo(𝒞) Qu Qu𝕃L Qu Qu𝕃CC AARAA Ho(𝒟) \mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})
Proof

This follows immediately from the fact that passing to homotopy categories of model categories is a double pseudofunctor from the double category of model categories to the double category of squares in Cat (Prop. ).

Example

(Quillen adjoint triple from left and right Quillen functor)

Given an adjoint triple (Remark )

𝒞AAAAL AAAAC AAAAR𝒟 \mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{R}{\longrightarrow} } \mathcal{D}

such that CC is both a left Quillen functor as well as a right Quillen functor (Def. ) for given model category-structures on the categories 𝒞\mathcal{C} and 𝒟\mathcal{D}. Then this is a Quillen adjoint triple (Def. ) of the form

𝒞 Qu QuL𝒟 \mathcal{C} \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 Qu QuRC𝒟 \mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
Proof

The condition of a Quillen equivalence (99) is trivially satisfied (by Prop. ). Similarly the required 2-morphism (100)

𝒞 ACA 𝒟 C id id 𝒟 AAidAA 𝒟 \array{ \mathcal{C} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{D} &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{D} }

exists trivially. To check that its derived natural transformation (Def. ) is a natural isomorphism we need to check (by Prop. ) that for every fibrant and cofibrant object d𝒟d \in \mathcal{D} the composite

QC(d)p C(d)C(d)j C(d)PC(C) Q C(d) \overset{ p_{C(d)} }{\longrightarrow} C(d) \overset{ j_{C(d)} }{\longrightarrow} P C(C)

is a weak equivalence. But this is trivially the case, by definition of fibrant resolution/cofibrant resolution (Def. ; in fact, since CC is assumed to be both left and right Quillen, also C(d)C(d) is a fibrant and cofibrant objects and hence we may even take both p C(d)p_{C(d)} as well as j C(d)j_{C(d)} to be the identity morphism).

The following is the analog in homotopy theory of the adjoint triple of the adjoint triple colimit/constant functor/limit (Def. ):

Example

(Quillen adjoint triple of homotopy limits/colimits of simplicial sets)

Let 𝒞\mathcal{C} be a small category (Def. ), and write [𝒞 op,sSet Qu] proj/inj[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} for the projective/injective model structure on simplicial presheaves over 𝒞\mathcal{C} (Prop. ), which participate in a Quillen equivalence of the form

[𝒞 op,sSet Qu] inj QuAAidAAAAidAA[𝒞 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}

Moreover, the constant diagram-assigning functor

[𝒞 op,sSet]constsSet [\mathcal{C}^{op}, sSet] \overset{const}{\longleftarrow} sSet

is clearly a left Quillen functor for the injective model structure, and a right Quillen functor for the projective model structure.

Together this means that in the double category of model categories (Def. ) we have a 2-morphism of the form

sSet Qu const [𝒞 op,sSet Qu] proj const id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] inj \array{ sSet_{Qu} &\overset{const}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{const}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} }

Moreover, the derived natural transformation Ho(id)Ho(id) (Prop. ) of this square is invertible, if for every Kan complex XX

QconstXconstXPconstX Q const X \overset{}{\longrightarrow} const X \longrightarrow P const X

is a weak homotopy equivalence (by Prop. ), which here is trivially the case.

Therefore we have a Quillen adjoint triple (Def. ) of the form

[𝒞 op,sSet Qu] proj Qu QulimsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}
[𝒞 op,sSet Qu] inj Qu QulimconstsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{const}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}

The induced adjoint triple of derived functors on the homotopy categories (via Prop. ) is the homotopy colimit/homotopy limit adjoint triple

Ho([𝒞 op,sSet])AA𝕃limAA AAconstAA AAlimAAHo(sSet) Ho([\mathcal{C}^{op}, sSet]) \; \array{ \overset{\phantom{AA}\mathbb{L}\underset{\longrightarrow}{\lim}\phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}const\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}\underset{\longleftarrow}{\lim}\phantom{AA}}{\longrightarrow} } \; Ho(sSet)

More generally:

Example

(Quillen adjoint triple of homotopy Kan extension of simplicial presheaves)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories (Def. ), and let

𝒞AAFAA𝒟 \mathcal{C} \overset{\phantom{AA}F\phantom{AA}}{\longrightarrow} \mathcal{D}

be a functor between them. By Kan extension (Prop. ) enriched over sSet (Example ) this induces an adjoint triple between categories of simplicial presheaves (Def. ):

[𝒞 op,sSet]AAF !AA AAF *AA AAF *AA[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot}{ \phantom{AA}F_!\phantom{AA} }{\longrightarrow} \\ \underoverset{\bot}{ \phantom{AA}F^\ast\phantom{AA} }{\longleftarrow} \\ \overset{ \phantom{AA}F_\ast\phantom{AA} }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

where

F *XX(F()) F^\ast \mathbf{X} \;\coloneqq\; \mathbf{X}(F(-))

is the operation of precomposition with FF. This means that F *F^\ast preserves all objectwise cofibrations/fibrations/weak equivalences in the model structure on simplicial presheaves (Prop. ). Hence it is

  1. a right Quillen functor (Def. ) [𝒟 op,sSet] projF *[𝒞 op,sSet Qu] proj[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj};

  2. a left Quillen functor (Def. ) [𝒟 op,sSet] injF *[𝒞 op,sSet Qu] inj[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj};

and since

[𝒟 op,sSet] injAAAAidid[𝒟 op,sSet] proj [\mathcal{D}^{op}, sSet]_{inj} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} [\mathcal{D}^{op}, sSet]_{proj}

is also a Quillen adjunction (Def. ), these imply that F *F^\ast is also

  • a right Quillen functor [𝒟 op,sSet] injF *[𝒞 op,sSet Qu] proj[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}.

  • a left Quillen functor [𝒟 op,sSet] projF *[𝒞 op,sSet Qu] inj[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}.

In summary this means that we have 2-morphisms in the double category of model categories (Def. ) of the following form:

[𝒟 op,sSet Qu] proj AAF *AA [𝒞 op,sSet Qu] proj F * id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] injAAAandAAA[𝒟 op,sSet Qu] inj AAF *AA [𝒞 op,sSet Qu] proj F * id id [𝒞 op,sSet Qu] inj id [𝒞 op,sSet Qu] proj \array{ [\mathcal{D}^{op}, sSet_{Qu}]_{proj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} } \phantom{AAA} \text{and} \phantom{AAA} \array{ [\mathcal{D}^{op}, sSet_{Qu}]_{inj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} }

To check that the corresponding derived natural transformations Ho(id)Ho(id) are natural isomorphisms, we need to check (by Prop. ) that the composites

Q injF *Xp F *XF *Xj F *XP projF *X Q_{inj} F^\ast \mathbf{X} \overset{ p_{F^\ast \mathbf{X}} }{\longrightarrow} F^\ast \mathbf{X} \overset{ j_{F^\ast \mathbf{X}} }{\longrightarrow} P_{proj} F^\ast \mathbf{X}

are invertible in the homotopy category Ho([𝒞 op,sSet Qu] inj/proj)Ho([\mathcal{C}^{op}, sSet_{Qu}]_{inj/proj}) (Def. ), for all fibrant-cofibrant simplicial presheaves X\mathbf{X} in [𝒞 op,sSet Qu] proj/inj[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution (Def. ).

Hence we have a Quillen adjoint triple (Def. ) of the form

(101)[𝒞 op,sSet Qu] proj/injAAF !AA AAF *AA AAF *AA[𝒟 op,sSet Qu] projAAAandAAA[𝒞 op,sSet Qu] proj/injAAF !AA AAF *AA AAF *AA[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \phantom{AAA} \text{and} \phantom{AAA} [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

The corresponding derived adjoint triple on homotopy categories (Prop. ) is that of homotopy Kan extension:

Ho([𝒞 op,sSet])A aA𝕃F !A aA A aAF *𝕃F *A AA aF *AHo([𝒟 op,sSet]) Ho([\mathcal{C}^{op}, sSet]) \array{ \underoverset{\bot \phantom{\simeq A_a}}{ \phantom{A}\mathbb{L}F_! \phantom{\simeq A_a}\phantom{A} }{\longrightarrow} \\ \underoverset{\phantom{\simeq A_a} \bot}{ \phantom{A}\mathbb{R}F^\ast \simeq \mathbb{L}F^\ast\phantom{A} }{\longleftarrow} \\ \overset{ \phantom{A} \phantom{A_a \simeq} \mathbb{R}F_\ast\phantom{A} }{\longrightarrow} } Ho([\mathcal{D}^{op}, sSet])
Example

(Quillen adjoint quadruple of homotopy Kan extension of simplicial presheaves along adjoint pair)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories (Def. ), and let

𝒞AARAAAALAA𝒟 \mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longrightarrow}} {\bot} \mathcal{D}

be a pair of adjoint functors (Def. ). By Kan extension this induces an adjoint quadruple (Prop. ) between categories of simplicial presheaves (Def. )

[𝒞 op,sSet]A aL !A a L *R ! A aL *R * A aR *[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a}}{ L_! \phantom{\simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot }{ L^\ast \simeq R_! }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot}{ L_\ast \simeq R^\ast }{\longrightarrow} \\ \overset{ \phantom{A_a \simeq } R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

By Example the top three as well as the bottom three of these form Quillen adjoint triples (Def. ) for model structures on simplicial presheaves (Prop. ) in two ways (101). If for the top three we choose the first version, and for the bottom three the second version from (101), then these combine to a Quillen adjoint quadruple of the form

[𝒞 op,sSet Qu] proj Qu QuL !=A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{= A_a}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] inj Qu QuL *=R ![𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast = R_!}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] inj Qu QuA=R *L *=R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{A=} R_\ast}{\longleftarrow}} {\overset{L_\ast = R^\ast}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}
Example

(Quillen adjoint quintuple of homotopy Kan extension of simplicial presheaves along adjoint triple)

Let 𝒞\mathcal{C} and 𝒟\mathcal{D} be small categories (Def. ) and let

𝒞AAAAL AAAAC AAAAC𝒟 \mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} } \mathcal{D}

be a triple of adjoint functors (Remark ). By Kan extension (Prop. ) enriched over sSet (Def. ) this induces an adjoint quintuple between categories of simplicial presheaves

(102)[𝒞 op,sSet]A aA aL !A aA a A aL *C !A a A aL *C *R ! A aA aA aC *R * A aA aA aC *R *[𝒟 op,sSet] [\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a \simeq A_a}}{ L_! \phantom{\simeq A_a \simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot \phantom{\simeq A_a} }{ L^\ast \simeq C_! \phantom{\simeq A_a} }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot \phantom{\simeq} \bot}{ L_\ast \simeq C^\ast \simeq R_! }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \bot}{ \phantom{A_a \simeq } C_\ast \simeq R^\ast }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \phantom{\bot}}{ \phantom{A_a \simeq C_\ast \simeq} R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]

By Example the top four functors in (102) form a Quillen adjoint quadruple (Def. ) on model structures on simplicial presheaves (Prop. ) ending in a right Quillen functor

[𝒞 op,sSet Qu] injC *R *[𝒞 op,sSet Qu] inj. [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \overset{C_\ast \simeq R^\ast}{\longrightarrow} [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \,.

But R *R^\ast here is also a left Quillen functor (as in Example ), and hence this continues by one more Quillen adjoint triple via Example to a Quillen adjoint quintuple of the form

[𝒞 op,sSet Qu] projA Qu QuAL !A aA a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C !A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C *R ![𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}
[𝒞 op,sSet Qu] injA Qu QuAA aA aR *A aC *R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

Alternatively, we may regard the bottom four functors in (102) as a Quillen adjoint quadruple via example , whose top functor is then the left Quillen functor

[𝒞 op,sSet Qu] projL *[𝒟 op,sSet Qu] proj. [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \overset{ L^\ast }{\longleftarrow} [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \,.

But this is also a right Quillen functor (as in Example ) and hence we may continue by one more Quillen adjoint triple upwards (via Example ) to obtain a Quillen adjoint quintuple, now of the form

[𝒞 op,sSet Qu] projA Qu QuAL !A aA a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] projA Qu QuAL *C !A a[𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAL *C *R ![𝒟 op,sSet Qu] proj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}
[𝒞 op,sSet Qu] injA Qu QuAA aA aR *A aC *R *[𝒟 op,sSet Qu] inj [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}

We now discuss how to extract derived adjoint modalities from systems of Quillen adjoint triples. First we consider some preliminary lemmas.

Lemma

(derived adjunction units of Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

such that the two model structures 𝒞 1\mathcal{C}_1 and 𝒞 2\mathcal{C}_2 on the category 𝒞\mathcal{C} share the same class of weak equivalences.

Then:

  1. the derived adjunction unit of (LC)(L \dashv C) in 𝒞 1\mathcal{C}_1 (Def. ) differs only by a weak equivalence from the plain adjunction unit (Def. ).

  2. the derived adjunction counit of (CR)(C \dashv R) (Def. ) differs only by a weak equivalence form the plain adjunction counit (Def. ).

Proof

By Def. , the derived adjunction unit is on cofibrant objects c𝒞 1c \in \mathcal{C}_1 given by

cη cCL(c)C(j L(c))CPL(c) c \overset{\eta_c}{\longrightarrow} C L (c) \overset{ C(j_{L(c)}) }{\longrightarrow} C P L (c)

Here the fibrant resolution-morphism j P(c)j_{P(c)} is an acyclic cofibration in 𝒟\mathcal{D}. Since CC is also a left Quillen functor 𝒟C𝒞 2\mathcal{D} \overset{C}{\to} \mathcal{C}_2, the comparison morphism C(j L(c))C(j_{L(c)}) is an acyclic cofibration in 𝒞 2\mathcal{C}_2, hence in particular a weak equivalence in 𝒞 2\mathcal{C}_2 and therefore, by assumption, also in 𝒞 1\mathcal{C}_1.

The derived adjunction counit of the second adjunction is

CQR(c)C(p R(c))CR(c)ϵ cc C Q R (c) \overset{ C(p_{R(c)}) }{\longrightarrow} C R (c) \overset{ \epsilon_c }{\longrightarrow} c

Here the cofibrant resolution-morphisms p R(c)p_{R(c)} is an acyclic fibration in 𝒟\mathcal{D}. Since CC is also a right Quillen functor 𝒟C𝒞 1\mathcal{D} \overset{C}{\to} \mathcal{C}_1, the comparison morphism C(p R(c))C(p_{R(c)}) is an acyclic fibration in 𝒞 1\mathcal{C}_1, hence in particular a weak equivalence there, hence, by assumption, also a weak equivalence in 𝒞 2\mathcal{C}_2.

Lemma

(fully faithful functors in Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

𝒞 1 Qu QuL𝒟 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}
𝒞 2 Qu QuRC𝒟 \mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}

If LL and RR are fully faithful functors (necessarily jointly, by Prop. ), then so are their derived functors 𝕃L\mathbb{L}L and R\mathbb{R}R (Prop. ).

Proof

We discuss that RR being fully faithful implies that R\mathbb{R}R is fully faithful. Since also the derived functors form an adjoint triple (by Prop. ), this will imply the claim also for LL and 𝕃L\mathbb{L}L, by Prop. .

By Lemma the derived adjunction counit of CRC \dashv R is, up to weak equivalence, the ordinary adjunction counit. But the latter is an isomorphism, since RR is fully faithful (by this Prop.). In summary this means that the derived adjunction unit of (CR)(C \dashv R) is a weak equivalence, hence that its image in the homotopy category is an isomorphism. But the latter is the ordinary adjunction unit of 𝕃CR\mathbb{L}C \dashv \mathbb{R}R (by this Prop.), and hence the claim follows again by that Prop..

Lemma

(fully faithful functors in Quillen adjoint quadruple)

Given a Quillen adjoint quadruple (Def. )

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

if any of the four functors is fully faithful functor, then so is its derived functor.

Proof

Observing that each of the four functors is either the leftmost or the rightmost adjoint in the top or the bottom adjoint triple within the adjoint quadruple, the claim follows by Lemma .

In summary:

Proposition

(derived adjoint modalities from fully faithful Quillen adjoint quadruples)

Given a Quillen adjoint quadruple (Def. )

𝒞 1 Qu QuL=A𝒟 1 \mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuC=L𝒟 1 \mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1
𝒞 2 Qu QuA=RR=C𝒟 2 \mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2

then the corresponding derived functors form an adjoint quadruple

Ho(𝒞)AAAA𝕃L AAAA𝕃CC𝕃L AAAAR𝕃CC AAAARHo(𝒟) Ho(\mathcal{C}) \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}L }{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}C \simeq \mathbb{R}C \simeq \mathbb{L}L' }{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{R}R \simeq \mathbb{L}C' \simeq \mathbb{R}C' }{\longrightarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{ \mathbb{R}R' }{\longleftarrow} } Ho(\mathcal{D})

Moreover, if one of the functors in the Quillen adjoint quadruple is a fully faithful functor, then so is the corresponding derived functor.

Hence if the original adjoint quadruple induces an adjoint modality on 𝒞\mathcal{C} (Def. )

\bigcirc \dashv \Box \dashv \lozenge

or on 𝒟\mathcal{D}

\Box \dashv \bigcirc \dashv \triangle

then so do the corresponding derived functors on the homotopy categories, respectively.

Proof

The existence of the derived adjoint quadruple followy by Prop. and by uniqueness of adjoints (this Prop.).

The statement about fully faithful functors is Lemma . The reformulation in terms of adjoint modalities is by this Prop.

\,

\infty-Toposes

The characterization of sheaf toposes as the left exact reflective localizations of presheaf toposes (Prop. ) now has an immediate generalization from the realm of locally presentable categories to that of combinatorial model categories and their corresponding locally presentable (∞,1)-categories (Def. ): This yields concept of model toposes and (∞,1)-toposes (Def. below).

\,

Definition

(model topos and (∞,1)-topos)

A combinatorial model category (Def. ) is a model topos if it has a presentation via Dugger's theorem (Prop. )

(103)[𝒞 op,sSet Qu] proj,S QuAAidAAid[𝒞 op,sSet Qu] projCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{proj,S} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \;\; \in CombModCat

such that the left derived functor 𝕃id\mathbb{L}id preserves finite homotopy limits.

We denote the image of such a combinatorial model category under the localization functor γ\gamma in Ho(CombModCat) (Def. ) by

Sh (𝒞)γ([𝒞 op,sSet Qu] proj,S)Ho(CombModCat) Sh_\infty(\mathcal{C}) \;\coloneqq\; \gamma([\mathcal{C}^{op}, sSet_{Qu}]_{proj,S}) \;\in\; Ho(CombModCat)

and call this an (∞,1)-topos over a site 𝒞\mathcal{C}. Moreover, we denote the image of the defining Quillen adjunction (103) in Ho(CombModCat) by

Sh (𝒞)AAAAlexPSh (𝒞)Ho(CombModCat). Sh_\infty(\mathcal{C}) \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{lex}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat) \,.

The following construction generalizes the Cech groupoid (Example ) as groupoids are generalized to Kan complexes (Def. ):

Example

(Cech nerve)

Let 𝒞\mathcal{C} be a site (Def. ). Then for every object X𝒞X \in \mathcal{C} and every covering {U iι iX}\{U_i \overset{\iota_i}{\to}X\} there is a simplicial presheaf (Example )

C({U i})[𝒞 op,sSet] C(\{U_i\}) \;\in\; [\mathcal{C}^{op}, sSet]

which in degree kk is given by the disjoint union of the kk-fold fiber products of presheaves over y(X)y(X) of the patches y(U i)[𝒞 op,Set]y(U_i) \in [\mathcal{C}^{op}, Set] of the cover, regarded as presheaves under the Yoneda embedding (Prop. )

C({U i}) ki 1,,i ky(U i 1)× y(X)y(U i 2)× y(X)× y(X)y(U i k). C(\{U_i\})_k \;\coloneqq\; \underset{i_1, \cdots, i_k}{\coprod} y(U_{i_1}) \times_{y(X)} y(U_{i_2}) \times_{y(X)} \cdots \times_{y(X)} y(U_{i_k}) \,.

The face maps are the evident projection morphisms, and the degeneracy maps the evident diagonal morphisms.

This is called the Cech nerve of the given cover.

By the definition of fiber products there is a canonical morphism of simplicial presheaves from the Cech nerve to y(X)y(X)

(104)C({U i})p {U i}y(X) C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X)

We call this the Cech nerve projection.

More generally, for

YfX[𝒞 op,Set] \mathbf{Y} \overset{f}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, Set]

any morphism of presheaves, there is the correspnding Cech nerve simplicial presheaf

C(f)[𝒞 op,sSet] C(f) \in [\mathcal{C}^{op}, sSet]

which in degree kk is the kk-fold fiber product of ff with itself:

C(f) kY× X× XYkfactors. C(f)_k \;\coloneqq\; \underset{ k \; \text{factors} }{ \underbrace{ \mathbf{Y} \times_{\mathbf{X}} \cdots \times_{\mathbf{X}} \mathbf{Y} }} \,.

The following is the generalization of Prop. , saying that Cech nerves are codescent-objects for (∞,1)-sheaves:

Proposition

(topological localization)

Let 𝒞\mathcal{C} be a site (Def. ) and let

S{C({U i})p {U i}y(X)|{U iι iX} icovering}Mor [𝒞 op,sSet] S \;\coloneqq\; \big\{ C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X) \;\vert\; \{U_i \overset{\iota_i}{\to} X\}_i \; \text{covering} \big\} \subset Mor_{[\mathcal{C}^{op}, sSet]}

be the set of projections (104) out of the Cech nerves (Example ) for coverings of all objects in the site, as a subset of the class of morphisms of simplicial presheaves over 𝒞\mathcal{C} (Example ).

Then the left Bousfield localization (Def. ) of the projective or injective model structure on simplicial presheaves (Prop. ), to be denoted

[𝒞 op,sSet Qu] proj/injloc QuAAidAAid[𝒞 op,sSet Qu] proj/inj [\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop {loc}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}

and to be called the (projective or injective) local model structure on simplicial presheaves, is left exact, in that it exhibits a model topos according to Def. , hence in that its image in Ho(CombModCat) is an (∞,1)-topos

Sh (𝒞)AAιAAlexPSh (𝒞). Sh_\infty(\mathcal{C}) \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{lex}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{C}) \,.
Proposition

(Quillen equivalence between projective and injective topological localization)

Let 𝒞\mathcal{C} be a site (Def. ) and let

S{C({U i})p {U i}y(X)|{U iι iX} icovering}Mor [𝒞 op,sSet] S \;\coloneqq\; \big\{ C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X) \;\vert\; \{U_i \overset{\iota_i}{\to} X\}_i \; \text{covering} \big\} \subset Mor_{[\mathcal{C}^{op}, sSet]}

be the set of projections (104) out of the Cech nerves (Example ) for coverings of all objects in the site, as a subset of the class of morphisms of simplicial presheaves over 𝒞\mathcal{C} (Example ).

If each Cech nerve C({U i})C(\{U_i\}) is already a cofibrant object in the projective model structure on simplicial presheaves (prop. ) then the identity functors constitute a Quillen equivalence (Def. ) between the corresponding topological localizations (Def. ) of the projective and the injective model structure on simplicial presheaves:

[𝒞 op,sSet Qu] injloc Qu Quidid[𝒞 op,sSet Qu] projloc [\mathcal{C}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}
Proof

First to see that we have a Quillen adjunction (Def. ): By Prop. this is the case before left Bousfield localization. By the nature of left Bousfield localization, and since the model structures are left proper simplicial model categories (by Prop. ), by Prop. it is sufficient to check that the right Quillen functor preserves fibrant objects. By Prop. this means to check that it preserves SS-local objects. But since C({U i})C(\{U_i\}) is assumed to be projectively cofibrant, and since injectively fibrant objects are already projectively fibrant, the condition on an injectively local object according to Def. is exactly the same as for a projectively local object.

Now to see that this Quillen adjunction is a Quillen equivalence, it is sufficient to check that the corresponding left/right derived functors induce an equivalence of categories on homotopy categories. By Prop. this is the case before left Bousfield localization. By Prop. it is thus sufficient to check that derived functors (before localization) preserves SS-local objects. By Prop. for this it is sufficient that the Quillen functors themselves preserve local objects. For the right Quillen functor we have just seen this in the previous paragaraph, for the left Quillen functor it follows analogously.

Example

(homotopy localization at 𝔸 1\mathbb{A}^1 over the site of 𝔸 n\mathbb{A}^ns)

Let 𝒞\mathcal{C} be any site (Def. ), and write [𝒞 op,sSet Qu] proj,loc[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc} for its local projective model category of simplicial presheaves (Prop. ).

Assume that 𝒞\mathcal{C} contains an object 𝔸𝒞\mathbb{A} \in \mathcal{C}, such that every other object is a finite product 𝔸 n𝔸××𝔸nfactors\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}, for some nn \in \mathbb{N}. (In other words, assume that 𝒞\mathcal{C} is also the syntactic category of Lawvere theory.)

Consider the 𝔸 1\mathbb{A}^1-homotopy localization (Def. ) of the (∞,1)-sheaf (∞,1)-topos over 𝒞\mathcal{C} (Prop. )

Sh (𝒞) 𝔸AAιAAL 𝔸Sh (𝒞)Ho(CombModCat) Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)

hence the left Bousfield localization of model categories

[𝒞 op,sSet Qu] proj,loc,𝔸 Qu QuAAidAAid[𝒞 op,sSet Qu] proj,locCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc} \;\; \in CombModCat

at the set of morphisms

S{𝔸 n×𝔸p 1𝔸 n} S \;\coloneqq\; \big\{ \mathbb{A}^n \times \mathbb{A} \overset{p_1}{\longrightarrow} \mathbb{A}^n \big\}

(according to Prop. ).

Then this is equivalent (Def. ) to ∞Grpd (Def. ),

GrpdSh (𝒞) 𝔸AAιAAL 𝔸Sh (𝒞)Ho(CombModCat) \infty Grpd \;\simeq\; Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)

in that the (constant functor \dashv limit)-adjunction (Def. )

(105)[𝒞 op,sSet Qu] inj,loc,𝔸limAAconstAAsSet QuCombModCat [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu} \;\;\;\; \in CombModCat

is a Quillen equivalence (Def. ).

Proof

First to see that (105) is a Quillen adjunction (Def. ): Since we have a simplicial Quillen adjunction before localization

[𝒞 op,sSet Qu] injlimAAconstAAsSet Qu [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu}

(by Example ) and since both model categories here are left proper simplicial model categories (by Prop. and Prop. ), and since left Bousfield localization does not change the class of cofibrations (by Def. ) it is sufficient to show that lim\underset{\longleftarrow}{\lim} preserves fibrant objects (by Prop. ).

But by assumption 𝒞\mathcal{C} has a terminal object *=𝔸 0\ast = \mathbb{A}^0 (Def. ), which is hence the initial object of 𝒞 op\mathcal{C}^{op}, so that the limit operation is given just by evaluation on that object:

limX=X(𝔸 0). \underset{\longleftarrow}{\lim} \mathbf{X} \;=\; \mathbf{X}(\mathbb{A}^0) \,.

Hence it is sufficient to see that an injectively fibrant simplicial presheaf X\mathbf{X} is objectwise a Kan complex. This is indeed the case, by Prop. .

To check that (105) is actually a Quillen equivalence (Def. ), we check that the derived adjunction unit and derived adjunction counit (Def. ) are weak equivalences:

For XsSetX \in sSet any simplicial set (necessarily cofibrant), the derived adjunction unit is

Xid Xconst(X)(𝔸 0)const(j X)(𝔸 0)const(PX)(𝔸 0) X \overset{id_X}{\longrightarrow} const(X)(\mathbb{A}^0) \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow} const(P X)(\mathbb{A}^0)

where Xj XPXX \overset{j_X}{\longrightarrow} P X is a fibrant replacement (Def. ). But const()(𝔸 0)const(-)(\mathbb{A}^0) is clearly the identity functor and the plain adjunction unit is the identity morphism, so that this composite is just j Xj_X itself, which is indeed a weak equivalence.

For the other case, let X[𝒞 op,sSet Qu] inj,loc,𝔸 1\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1} be fibrant. This means (by Prop. ) that X\mathbf{X} is fibrant in the injective model structure on simplicial presheaves as well as in the local model structure, and is a derived-𝔸 1\mathbb{A}^1-local object (Def. ), in that the derived hom-functor out of any 𝔸 n×𝔸 1p 1𝔸 n\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n into X\mathbf{X} is a weak homotopy equivalence:

Hom(p 1):Hom(𝔸 n,X)WHom(𝔸 n×𝔸 1,X) \mathbb{R}Hom( p_1 ) \;\colon\; \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X}) \overset{\in W}{\longrightarrow} \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X})

But since X\mathbf{X} is fibrant, this derived hom is equivalent to the ordinary hom-functor (Lemma ), and hence with the Yoneda lemma (Prop. ) we have that

X(p 1):X(𝔸 n)WX(𝔸 n+1) \mathbf{X}(p_1) \;\colon\; \mathbf{X}(\mathbb{A}^n) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^{n+1})

is a weak equivalence, for all nn \in \mathbb{N}. By induction on nn this means that in fact

X(𝔸 0)WX(𝔸 n) \mathbf{X}(\mathbb{A}^0) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^n)

is a weak equivalence for all nn \in \mathbb{N}. But these are just the components of the adjunction counit

const(X(𝔸 0))WϵX const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}

which is hence also a weak equivalence. Hence for the derived adjunction counit

const(QX)(𝔸 0)const(p X(𝔸 0))const(X(𝔸 0))WϵX const (Q \mathbf{X})(\mathbb{A}^0) \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow} const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}

to be a weak equivalence, it is now sufficient to see that the value of a cofibrant replacement p Xp_{\mathbf{X}} on 𝔸 0\mathbb{A}^0 is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.

Proposition

(Cech nerve-projection of local epimorphism is local weak equivalence)

Let 𝒞\mathcal{C} be a site (Def. ) and let

YAfAX[𝒞 op,Set] \mathbf{Y} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, Set]

be a local epimorphism (Def. ) in its category of presheaves. Then the corresponding Cech nerve-projection (Def. )

C(f)X[𝒞 op,sSet Qu] proj,loc C(f) \overset{}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc}

is a weak equivalence in the local projective model structure on simplicial presheaves (Prop. ).

(Dugger-Hollander-Saksen 02, corollary A.3)

\,

Gros \infty-Toposes

We have established above enough higher category theory/homotopy theory that it is now fairly straightforward to generalize the discussion of gros toposes to model toposes/(∞,1)-toposes.

\,

Cohesive \infty-Toposes

The following is a refinement to homotopy theory of the notion of cohesive topos (Def. ):

Definition

(cohesive model topos)

An (∞,1)-topos H\mathbf{H} (Def. ) is called a cohesive (∞,1)-topos if it is presented by a model topos [𝒞 op,sSet Qu] loc[\mathcal{C}^{op}, sSet_{Qu}]_{loc} (Def. ) which admits a Quillen adjoint quadruple (Def. ) to the classical model category of simplicial sets (Def. ) of the form

[𝒞 op,sSet Qu] proj/inj A QuAAΠAA A QuADiscA A QuAAΓAA AcoDiscA sSet Qu \array{ [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA}\Pi\phantom{AA}}{\longrightarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu} }{\phantom{A}Disc\phantom{A}}{\hookleftarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow} \\ \overset{\phantom{A}coDisc\phantom{A}}{\hookleftarrow} \\ } sSet_{Qu} }

such that

  1. (DiscΓ)(Disc \dashv \Gamma) is a Quillen coreflection (Def. );

  2. (ΓcoDisc)(\Gamma \dashv coDisc) is a Quillen reflection (Def. );

  3. Π\Pi preserves finite products.

The following is the analog of Example :

Example

(Quillen adjoint quadruple on simplicial presheaves over site with finite products)

Let 𝒞\mathcal{C} be a small category (Def. ) with finite products (hence with a terminal object *𝒞\ast \in \mathcal{C} and for any two objects X,Y𝒞X,Y \in \mathcal{C} their Cartesian product X×Y𝒞X \times Y \in \mathcal{C}). By Example the terminal object is witnessed by an adjunction

(106)*AAAA𝒞 \ast \underoverset {\underset{}{\hookrightarrow}} {\overset{\phantom{AAAA}}{\longleftarrow}} {\bot} \mathcal{C}

Consider the category of simplicial presheaves [𝒞 op,sSet][\mathcal{C}^{op}, sSet] (Example ) with its projective and injective model structure on simplicial presheaves (Prop. ).

Then Kan extension (Prop. ) enriched over sSet (Example ) along the adjoint pair (106) yields a simplicial Quillen adjoint quadruple (Def. )

(107)[𝒞 op,Set Qu] proj/inj A QuAAAΠAAA A QuAADiscAA A QuAAAΓAAA AAcoDiscAAsSet Qu [\mathcal{C}^{op}, Set_{Qu}]_{proj/inj} \array{ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } sSet_{Qu}

such that:

  1. the functor Γ\Gamma sends a simplicial presheaf Y\mathbf{Y} to its simplicial set of global sections, which here is its value on the terminal object:

    (108)ΓY =lim𝒞Y Y(*) \begin{aligned} \Gamma \mathbf{Y} & = \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \mathbf{Y} \\ & \simeq \mathbf{Y}(\ast) \end{aligned}
  2. (DiscΓ)(Disc \dashv \Gamma) is a Quillen coreflection (Def. )

  3. (ΓcoDisc)(\Gamma \dashv coDisc) is a Quillen reflection (Def. );

  4. Π\Pi preserves finite products:

Hence the category of simplicial presheaves over a small category with finite products is a cohesive (∞,1)-topos (Def. ).

Proof

The Quillen adjoint quadruple follows as the special case of Example applied to the adjoint pair

*𝒞 \ast \underoverset {\underset{}{\hookrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}

given by inclusion of the terminal object (Example ).

Since the plain adjoint quadruple has (ΠDisc)(\Pi \dashv \Disc) a reflective subcategory inclusion and (DiscΓ)(Disc \dashv \Gamma) a coreflective subcategory inclusion (Example ) the Quillen (co-)reflection follows by Prop.

The following is a refinement to homotopy theory of the notion of cohesive site (Def. ):

Definition

(∞-cohesive site)

We call a site 𝒞\mathcal{C} (Def. ) ∞-cohesive if the following conditions are satisfied:

  1. The category 𝒞\mathcal{C} has finite products;

  2. For every covering family {U iX} i\{U_i \to X\}_i in the given coverage on 𝒞\mathcal{C}, the induced Cech nerve simplicial presheaf (Example ) C({U i})[𝒞 op,sSet]C(\{U_i\}) \in [\mathcal{C}^{op}, sSet] satisfies the following conditions

    1. C({U i})C(\{U_i\}) is a cofibrant object in the projective model structure on simplicial presheaves [𝒞 op,sSet Qu] proj[\mathcal{C}^{op}, sSet_{Qu}]_{proj} (Prop. )

    2. The simplicial set obtained as the degreewise colimit over the Cech nerve is weakly homotopy equivalent to the point

      lim𝒞 opC({U i})* \underset{ \underset{ \mathcal{C}^{op} }{\longrightarrow} }{\lim} C(\{U_i\}) \simeq \ast
    3. The simplicial set obtained at the degreewise limit over the Cech nerve is weakly homotopy equivalent to the underlying set of points of XX:

      C({U i})𝒞 opHom 𝒞(*,X). \underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{C(\{U_i\})} \simeq Hom_{\mathcal{C}}(\ast, X) \,.

The following is the analog of Prop. :

Proposition

(model topos over ∞-cohesive site is cohesive model topos)

Let 𝒞\mathcal{C} be an ∞-cohesive site (Def. ). Then the (∞,1)-topos (Def. ) over it, obtained by topological localization (Prop. ) is a cohesive (∞,1)-topos (Def. ).

Proof

By Example we have the required Quillen adjoint quadruple on the projective model structure on simplicial presheaves, i.e. before left Bousfield localization at the Cech nerve projections

[𝒞 op,Set Qu]AAAΠAAA AADiscAA AAAΓAAA AAcoDiscAAsSet Qu [\mathcal{C}^{op}, Set_{Qu}] \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } sSet_{Qu}

Hence it remains to see that these Quillen adjunctions pass to the local model structures [𝒞 op,Set Qu] proj/inj,loc[\mathcal{C}^{op}, Set_{Qu}]_{proj/inj, loc} from Prop. , and that DiscDisc and coDisccoDisc then still participate in Quillen (co-)reflections.

By Prop. and Prop. all model structures involved are left proper simplicial model categories, and hence we may appeal to Prop. for recognition of the required Quillen adjunctions. Since, moreover, left Bousfield localization does not change the class of cofibrations (Def. ), this means that we are reduced to checking that all right Quillen functors in the above global Quillen adjoint quadruple preserve fibrant objects with respect to the local model structure.

For the Quillen adjunctions

(ΠDisc),(ΓcoDisc):[𝒞 op,sSet Qu] projsSet Qu (\Pi \dashv Disc), (\Gamma \dashv coDisc) \;\colon\; [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \leftrightarrow sSet_{Qu}

this means to check that for every Kan complex SsSetS \in sSet the simplicial presheaves Disc(S)Disc(S) and coDisc(S)coDisc(S) are derived-local objects (Def. , Prop. ) with respect to the Cech nerve projections. Since DiscDisc and coDisccoDisc are right Quillen functors with respect to the global model projective model structure, Disc(S)Disc(S) and coDisc(S)coDisc(S) are globally projectively fibrant simplicial presheaves. Since, moreover, C({U i})C(\{U_i\}) is projectively cofibrant by assumption, and since the representables X𝒞X \in \mathcal{C} are projectively cofibrant by Prop. , the value of the derived hom-functor reduces to that of the ordinary enriched hom-functor (Def. ), and hence the condition is that

[𝒞 op,sSet](X,Disc(S)) W [𝒞 op,sSet](C({U i}),Disc(S)) [𝒞 op,sSet](X,coDisc(S)) W [𝒞 op,sSet](C({U i}),coDisc(S)) \array{ [\mathcal{C}^{op}, sSet]( X, Disc(S) ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( C(\{U_i\}), Disc(S) ) \\ [\mathcal{C}^{op}, sSet]( X, coDisc(S) ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( C(\{U_i\}), coDisc(S) ) }

are weak equivalences. But now by the ordinary adjunction hom-isomorphism (10), these are identified with

[𝒞 op,sSet](limX,S) W [𝒞 op,sSet](limC({U i}),S) [𝒞 op,sSet](limX,S) W [𝒞 op,sSet](limC({U i}),S) \array{ [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim} X, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim}C(\{U_i\}), S ) \\ [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}X, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}C(\{U_i\}), S ) }

Since the colimit of a representable is the singleton (Lemma ) and since the limit over the opposite of a category with terming object is evaluation at that object, this in turn is equivalent to

[𝒞 op,sSet](*,S) W [𝒞 op,sSet](limC({U i}),S) [𝒞 op,sSet](Hom 𝒞(*,X),S) W [𝒞 op,sSet](limC({U i}),S) \array{ [\mathcal{C}^{op}, sSet]( \ast, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim}C(\{U_i\}), S ) \\ [\mathcal{C}^{op}, sSet]( Hom_{\mathcal{C}}(\ast, X), S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}C(\{U_i\}), S ) }

Here we recognize the internal hom in simplicial sets from the weak equivalences of the definition of an ∞-cohesive site (Def. ), which necessarily go between cofibrant simplicial sets, into a fibrant simplicial set SS. Hence this is the derived hom-functor (Def. ) in the classical model structure on simplicial sets. Since the latter is a simplicial model category (Def. ) by Prop. , these morphisms are indeed weak equivalences of simplicial sets.

This establishes that (ΠDisc)(\Pi \dashv Disc) and (ΓcoDisc)(\Gamma \dashv \coDisc) descent to Quillen adjunctions on the local model structure. Finally, it is immediate that Γ\Gamma preserves fibrant objects, and hence also (DiscΓ)(Disc \dashv \Gamma) passes to the local model structure.

The following is the analog in homotopy theory of the cohesive adjoint modalities from Def. :

Definition

(adjoint triple of derived adjoint modal operators on homotopy category of cohesive model topos)

Given a cohesive model topos (Def. ), its adjoint quadruple (Remark ) of derived functor between homotopy categorues (via Prop. )

(109)ΠDiscΓcoDisc:Ho([𝒞 op,sSet Qu] loc)AAAΠ 0AAA AADiscAA AAAΓAAA AAcoDiscAAHo(sSet) \Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Ho(sSet)

induce, by composition of functors, an adjoint triple (Remark ) of adjoint modalities (via Prop. ):

ʃ:Ho([𝒞 op,sSet Qu] loc)ʃDiscΠ 0 DiscΓ coDiscΓHo([𝒞 op,sSet Qu] loc). &#643; \dashv \flat \dashv \sharp \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \array{ \overset{ &#643; \;\coloneqq\; Disc \circ \Pi_0 }{\hookleftarrow} \\ \overset{\flat \;\coloneqq\; Disc \circ \Gamma }{\longrightarrow} \\ \overset{ \sharp \;\coloneqq\; coDisc\circ \Gamma }{\hookleftarrow} } Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \,.

Since DiscDisc and coDisccoDisc are fully faithful functors by assumption, these are (co-)modal operators (Def. ), (by Prop. and Prop. ).

We pronounce these as follows:

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ 0&#643; \;\coloneqq\; Disc \circ \Pi_0 A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

\,

Elastic \infty-Toposes

The following is a refinement to homotopy theory of the notion of elastic topos (Def. ):

Definition

(elastic model topos

Given a cohesive model topos [𝒞 red op,sSet Qu] proj/injloc[\mathcal{C}_{red}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc} (Def. ), a differentially cohesive or elastic model topos over it is another cohesive model topos [𝒞 red op,sSet Qu] proj/injloc[\mathcal{C}_{red}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc} equipped with a system of Quillen adjoint quadruples (Def. ) of the form

sSet Qu Qu QuΠ red[𝒞 red op,sSet Qu] projloc Qu Quidid[𝒞 red op,sSet Qu] projloc Qu Quι inf[𝒞 inf op,sSet Qu] projloc sSet Qu Qu QuΠ red[𝒞 red op,sSet Qu] projloc Qu Quid[𝒞 red op,sSet Qu] injrloc Qu QuΠ inf[𝒞 inf op,sSet Qu] projloc sSet Qu Qu QuDisc red[𝒞 red op,sSet Qu] injrloc Qu Quid[𝒞 inf op,sSet Qu] injrloc Qu QuDisc inf[𝒞 inf op,sSet Qu] injrloc sSet Qu Qu QucoDisc redΓ red[𝒞 red op,sSet Qu] injrloc Qu Quidid[𝒞 inf op,sSet Qu] injrloc Qu QuΓ inf[𝒞 inf op,sSet Qu] injrloc \array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc}} }

such that

  1. (ι infΠ inf)(\iota_{inf} \dashv \Pi_{inf}) is a Quillen coreflection (Def. );

  2. (Π infDisc inf)(\Pi_{inf} \dashv Disc_{inf}) is a Quillen reflection (Def. ).

Definition

(∞-elastic site)

For 𝒞 red\mathcal{C}_{red} an ∞-cohesive site (Def. ), an infinitesimal neighbourhood site of 𝒞 red\mathcal{C}_{red} is a coreflective subcategory-inclusion into another ∞-cohesive site 𝒞\mathcal{C}

𝒞 redAAΠ infAAι inf𝒞 \mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}

such that

  1. both ι inf\iota_{inf} and Π inf\Pi_{inf} send covers to covers;

  2. the left Kan extension of ι inf\iota_{inf} preserves fiber products y(U i)× y(X)y(u j)y(U_i) \times_{y(X)} y(u_j) of morphisms in a covering {U iι iX}\{U_i \overset{\iota_i}{\to} X\};

  3. if {U iι iX}\{ U_i \overset{\iota_i}{\to} X \} is a covering family in 𝒞 red\mathcal{C}_{red}, and p(X^)Xp(\widehat X) \longrightarrow X is any morphism in 𝒞 red\mathcal{C}_{red}, then there is a covering familiy {U^ iι^ jX^}\{ \widehat U_i \overset{\widehat\iota_j}{\to} \widehat X \} such that for all ii there is a jj and a commuting square of the form

    (110)Π inf(U^ j) U i Π inf(ι^ j) ι i Π inf(X^) X \array{ \Pi_{inf}(\widehat U_j) &\longrightarrow& U_i \\ {}^{\mathllap{ \Pi_{inf}(\widehat\iota_j) }}\Big\downarrow && \Big\downarrow{}^{\mathrlap{ \iota_i }} \\ \Pi_{inf}(\widehat X) &\longrightarrow& X }

We also call this an ∞-elastic site, for short.

Proposition

(model topos over ∞-elastic site is elastic model topos)

Let

𝒞 redAAΠ infAAι inf𝒞 \mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}

be an ∞-elastic site (Def. ). Then Kan extension (Prop. ) enriched over sSet (Example ) induces on the corresponding cohesive model toposes (Prop. ) the structure of an elastic model topos (Def. ).

Proof

By Example we have a Quillen adjoint quadruple for the global projective model structure on simplicial presheaves of the form

[𝒞 red op,sSet Qu] proj Qu Quι inf[𝒞 op,sSet Qu] proj [\mathcal{C}^{op}_{red}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{ \iota_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}
[𝒞 red op,sSet Qu] inj Qu QuC=L[𝒞 op,sSet Qu] proj [\mathcal{C}^{op}_{red}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}
[𝒞 red op,sSet Qu] inj Qu QuA=RR=C[𝒞 op,sSet Qu] inj [\mathcal{C}^{op}_{red}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}

Here we denote left Kan extension along a functor by the same symbol as that functor, which is consistent by Prop. .

By Prop. all model categories appearing here are left proper simplicial model categories, and by Def. left Bousfield localization retains the class of cofibrations. Therefore Prop. says that to see that this is also a Quillen adjoint quadruple for the local model structure on simplicial presheaves (Prop. ) it is sufficient that, for each Quillen adjunction, the right adjoint preserves fibrant objects, hence Cech-local objects (Def. ).

For each right adjoint RR here this means to consider any covering {U iX}\{U_i \overset{}{\to} X\} (either in 𝒞 red\mathcal{C}_{red} or in 𝒞\mathcal{C}) with induced Cech nerve C({U i})C(\{U_i\}) (Example ) and to check that for a fibrant object X\mathbf{X} in the global projective/injective model structure on simplicial presheaves, that

[X,RX][C({U i}),RX] [X, R\mathbf{X}] \longrightarrow [ C(\{U_i\}), R\mathbf{X} ]

is a weak equivalence. Notice that this is indeed already the image under the correct derived hom-functor, Def. , since both sites are assumed to be ∞-cohesive sites (Def. ), which means in particular that C({U i})C(\{U_i\}) is projectively cofibrant, and hence also injectively cofibrant, by Prop. .

Now by the enriched adjunction-isomorphism (47) this means equivalently that

(111)[LX,X][LC({U i}),X] [L X, \mathbf{X}] \longrightarrow [ L C(\{U_i\}), \mathbf{X} ]

is a weak equivalence. This we now check in each of the three cases:

For the case (ι infΠ inf)(\iota_{inf} \dashv \Pi_{inf}) we have that

ι infC({U i})C({ι infU i}) \iota_{inf} C(\{U_i\}) \simeq C(\{\iota_{inf} U_i\})

by the assumption that ι inf\iota_{inf} preserves fiber products of Yoneda embedding-images of morphisms in a covering. Moreover, by the assumption that ι inf\iota_{inf} preserves covering-families, C({ι infU i})C(\{\iota_{inf} U_i\}) is itself the Cech nerve of a covering family, and hence (111) is a weak equivalence since X\mathbf{X} is assumed to be a local object.

The same argument directly applies also to (Π infDisc inf)(\Pi_{inf} \dashv Disc_{inf}), where now the respect of Π inf\Pi_{inf} for fiber products follows already from the fact that this is a right adjoint (since right adjoints preserve limits, Prop. ).

In the same way, for (Disc infΓ inf)(Disc_{inf} \dashv \Gamma_{inf}) we need to check that [C({Disc infU i})Disc infX,X][ C(\{Disc_{inf}U_i\}) \to Disc_{inf} X, \mathbf{X} ] is a weak equivalence. Now Disc infDisc_{inf} is no longer a left Kan extension, hence Disc inf(U i)Disc inf(X)Disc_{inf}(U_i) \to Disc_{inf}(X) is no longer a morphism of representable presheaves. But the third assumption (110) on an \infty-elastic site manifestly means, under the adjunction isomorphism (10) for (Pi infDisc inf)(Pi_{inf} \dashv Disc_{inf}) that Disc inf(U i)Disc inf(X)Disc_{inf}(U_i) \to Disc_{inf}(X) is a local epimorphism (Def. ). Therefore Prop. implies that

C({Disc infU i})Disc infX C(\{Disc_{inf} U_i\}) \to Disc_{inf} X

is a weak equivalence. With this, the fact (Prop. with Prop. ) that [𝒞 op,sSet Qu] inj,loc[\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc} is a simplicial model category (Def. ) implies that [C({Disc infU i})Disc infX,X][C(\{Disc_{inf} U_i\}) \to Disc_{inf} X, \mathbf{X}] is a weak equivalence.

The following is a refinement to homotopy theory of the adjoint modalities on an elastic topos from Def. :

Definition

(derived adjoint modalities on elastic model topos)

Given an elastic model topos (def. ), composition composition of the derived functors (Prop. ) yields via Prop. and Prop. , the following adjoint modalities (Def. ) on the homotopy category (Def. )

&:Ho([𝒞 op,sSet] loc)ι infΠ inf Disc infΠ inf &Disc infΓ infHo([𝒞 op,sSet] loc). \Re \dashv \Im \dashv \& \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet]_{loc}) \array{ \overset{ \Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} }{\longleftarrow} \\ \overset{\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} }{\longrightarrow} \\ \overset{ \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} }{\longleftarrow} } Ho([\mathcal{C}^{op}, sSet]_{loc}) \,.

Since ι inf\iota_{inf} and Disc infDisc_{inf} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ^{op}, sSet]_{loc}) } and Prop. ).

We pronounce these as follows:

A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

Proposition

(progression of derived adjoint modalities on elastic model topos)

Let [𝒞 op,sSet] proj/injloc[\mathcal{C}^{op}, sSet]_{{proj/inj} \atop loc} be an elastic model topos (Def. ) and consider the corresponding derived adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ&#643; \;\coloneqq\; Disc \circ \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι infΠ inf\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} Disc infΠ inf\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} A\phantom{A}A\phantom{A} &Disc infΓ inf \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def.

& ʃ * \array{ \Re &\dashv& \Im &\dashv& \& \\ && \vee && \vee \\ && &#643; &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }

where we display also the bottom adjoint modality *\emptyset \dashv \ast (Example ), for completeness.

Proof

This is just as in Prop. .

\,

Solid \infty-Toposes

The following is a refinement to homotopy theory of the notion of solid topos (Def. ):

Definition

(solid model topos)

Given an elastic model topos [𝒞 inf op,sSet Qu] proj/injloc[\mathcal{C}^{op}_{inf}, sSet_{Qu}]_{{proj/inj} \atop loc} (Def. ) a solid model topos over it is another elastic model topos [𝒞 op,sSet Qu] proj/injloc[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc} and a system of Quillen adjoint quadruples (Def. ) as follows

sSet Qu Qu QuΠ red[𝒞 red op,sSet Qu] projloc Qu Quidid[𝒞 red op,sSet Qu] projloc Qu Quι inf[𝒞 inf op,sSet Qu] projloc Qu Quid[𝒞 inf op,sSet Qu] projloc Qu Queven[𝒞 op,sSet Qu] projloc sSet Qu Qu QuΠ red[𝒞 red op,sSet Qu] projloc Qu Quidid[𝒞 red op,sSet Qu] projloc Qu Quι inf[𝒞 inf op,sSet Qu] projloc Qu Quid[𝒞 inf op,sSet Qu] projloc Qu Quι sup[𝒞 op,sSet Qu] injrloc sSet Qu Qu QuΠ red[𝒞 red op,sSet Qu] projloc Qu Quid[𝒞 red op,sSet Qu] injrloc Qu QuΠ inf[𝒞 inf op,sSet Qu] projloc Qu Quid[𝒞 inf op,sSet Qu] injrloc Qu QuΠ sup[𝒞 op,sSet Qu] injrloc sSet Qu Qu QuDisc red[𝒞 red op,sSet Qu] injrloc Qu Quid[𝒞 inf op,sSet Qu] injrloc Qu QuDisc inf[𝒞 inf op,sSet Qu] injrloc Qu Quid[𝒞 inf op,sSet Qu] injrloc Qu QuDisc sup[𝒞 op,sSet Qu] injrloc sSet Qu Qu QucoDisc redΓ red[𝒞 red op,sSet Qu] injrloc Qu Quidid[𝒞 inf op,sSet Qu] injrloc Qu QuΓ inf[𝒞 inf op,sSet Qu] injrloc Qu Quid[𝒞 inf op,sSet Qu] projloc Qu QuΓ sup[𝒞 op,sSet Qu] projloc \array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{even}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc} \\ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\Pi_{sup}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\;\;id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{\;Disc_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc}} \underoverset {\underset{}{\phantom{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu} }} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc}} \underoverset {\phantom{{\longleftarrow}}} {\overset{\Gamma_{sup}}{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}} }

such that

  1. (evenι sup)(even \dashv \iota_{sup}) is a Quillen reflection (def. );

  2. (ι supΠ sup)(\iota_{sup} \dashv \Pi_{sup}) is a Quillen coreflection.

Definition

(∞-solid site)

For 𝒞 redAAΠ infAAι inf𝒞 inf \mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}_{inf} an ∞-elastic site (Def. ) over an ∞-cohesive site (Def. ), a super-infinitesimal neighbourhood site is a reflective/coreflective subcategory-inclusion into another ∞-elastic site 𝒞 redAAΠ infAAι inf𝒞 \mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}

*AAevenAA AAι infAA AAΠAA ADiscA𝒞 redAAevenAA AAι infAA AAΠ infAA ADiscA𝒞 infAAevenAA AAι supAA AAΠ supAA ADiscA𝒞 sup \ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } \mathcal{C}_{red} \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{inf} \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{sup}

such that

  1. all of eveneven, ι sup\iota_{sup} and Π inf\Pi_{inf} send covers to covers;

  2. the left Kan extension of eveneven preserves fiber products y(U i)× y(X)y(u j)y(U_i) \times_{y(X)} y(u_j) of morphisms in a covering {U iι iX}\{U_i \overset{\iota_i}{\to} X\};

Proposition

(model topos over ∞-solid site is solid model topos)

Let

*AAevenAA AAι infAA AAΠAA ADiscA𝒞 redAAevenAA AAι infAA AAΠ infAA ADiscA𝒞 infAAevenAA AAι supAA AAΠ supAA ADiscA𝒞 sup \ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } \mathcal{C}_{red} \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{inf} \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{sup}

be an ∞-solid site (Def. ). Then Kan extension (Prop. ) enriched over sSet (Example ) induces on the corresponding elastic model toposes (Prop. ) the structure of a solid model topos (Def. ).

The following is a refinement to homotopy theory of the modal operators on a solid topos from Def. :

Definition

(derived adjoint modalities on solid model topos)

Given a solid model topos [𝒞 op,sSet Qu] proj/injloc[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc} (Def. ), composition of derived functors via Prop. and Prop. , the following adjoint modalities (Def. )

Rh:Hι supeven ι supΠ sup RhDisc supΠ supH. \rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.

Since ι sup\iota_{sup} and Disc supDisc_{sup} are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. and Prop. ).

We pronounce these as follows:

A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

and we refer to the corresponding modal objects (Def. ) as follows:

  • a \rightsquigarrow-comodal object

    Xϵ X X \overset{\rightsquigarrow}{X} \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow} X

    is called a bosonic object;

  • a RhRh-modal object

    Xη X RhRhX X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X

    is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let [𝒞 op,sSet Qu] proj/injlco[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop lco} be a solid model topos (Def. ) and consider the adjoint modalities which it inherits

  1. for being a cohesive topos, from Def. ,

  2. for being an elastic topos, from Def. ,

  3. for being a solid topos, from Def. :

A\phantom{A} shape modality A\phantom{A}A\phantom{A} flat modality A\phantom{A}A\phantom{A} sharp modality A\phantom{A}
A\phantom{A} ʃDiscΠ&#643; \;\coloneqq\; Disc \Pi A\phantom{A}A\phantom{A} DiscΓ\flat \;\coloneqq\; Disc \circ \Gamma A\phantom{A}A\phantom{A} coDiscΓ\sharp \;\coloneqq\; coDisc \circ \Gamma A\phantom{A}
A\phantom{A} reduction modality A\phantom{A}A\phantom{A} infinitesimal shape modality A\phantom{A}A\phantom{A} infinitesimal flat modality A\phantom{A}
A\phantom{A} ι supι infΠ infΠ sup\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup} A\phantom{A}A\phantom{A} Disc supDisc infΠ infΠ sup\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup} A\phantom{A}A\phantom{A} &Disc supDisc infΓ infΓ sup \& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup} A\phantom{A}
A\phantom{A} fermionic modality A\phantom{A}A\phantom{A} bosonic modality A\phantom{A}A\phantom{A} rheonomy modality A\phantom{A}
A\phantom{A} ι supeven\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even A\phantom{A}A\phantom{A} ι supΠ sup\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} A\phantom{A}A\phantom{A} RhDisc supΠ sup Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} A\phantom{A}

Then these arrange into the following progression, via the preorder on modalities from Def. :

id id Rh & ʃ * \array{ id &\dashv& id \\ \vee && \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee && \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && &#643; &\dashv& \flat &\dashv& \sharp \\ && && && \vee && \vee \\ && && && \emptyset &\dashv& \ast }

where we are displaying, for completeness, also the adjoint modalities at the bottom *\emptyset \dashv \ast and the top ididid \dashv id (Example ).

Proof

This is just as in Prop. .

\,

(…)


  1. “the universality of the concept of adjointness, which was first isolated and named in the conceptual sphere of category theory” (Lawvere 69) “In all those areas where category theory is actively used the categorical concept of adjoint functor has come to play a key role.” (first line from An interview with William Lawvere, paraphrasing the first paragraph of Taking categories seriously)

Last revised on September 27, 2023 at 08:24:12. See the history of this page for a list of all contributions to it.