nLab
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 e.g. Freyd 65, Part II).

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 1YHom 𝒞(Y,X) Y \overset{f^{-1}}{\longrightarrow} Y \;\; \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 inft(X)C^\inft(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(d) f X(f)"=()f" d X(c) \array{ c && \mathbf{X}(d) \\ {}^{\mathllap{ f }}\big\downarrow && \big\uparrow{}^{\mathrlap{ \mathbf{X}(f) \, \text{"=}(-)\circ f\text{"}}} \\ d && \mathbf{X}(c) }

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 constiture 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 amorphism 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 morphism g:c 2c 1g \colon c_2 \to c_1 in 𝒞\mathcal{C} and for every morphisms 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

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

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 subsequence 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 \,.

If one of these has yet another adjoint in the other direction, we 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 LRL \dashv R (Def. ) provide, for each object XX in the domain of CC, a diagram

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

which is usefully thought of as exhibiting the nature of XX as being in between two opposite extreme aspects L(C(X))L(C(X)) and R(C(X))R(C(X)) 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 unit of DiscUDisc \dashv U and the adjunction counit 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 a 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}

of a pair of adjoint functors (Def. ), then application of the functor CC on any morphism XfY𝒞X \overset{f}{\to} Y \;\;\in \mathcal{C} is equal 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:

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

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:

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

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(CY,CY) ()˜ Hom 𝒞(Y,RCY) Hom 𝒟(C(f),CY) Hom 𝒞(f,RCY) Hom 𝒟(CX,CY) ()˜ Hom 𝒞(X,RCY)AAAAA{CYid CYCY} {Yη Y RCY} {CXC(f)CY} {Xη Y fRCY} \array{ Hom_{\mathcal{D}}( C Y , C Y ) &\overset{\widetilde {(-)}}{\longrightarrow}& Hom_{\mathcal{C}}( Y, R C Y ) \\ {}^{\mathllap{ Hom_{\mathcal{D}}(C(f), C Y) }} \big\downarrow && \!\!\!\!\! \big\downarrow^{\mathrlap{ Hom_{\mathcal{C}}( f, R C Y ) }} \\ Hom_{\mathcal{D}}( C X, C Y ) &\overset{\widetilde{ (-) }}{\longleftarrow}& Hom_{\mathcal{C}}( X, R C Y ) } \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 \} }

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 right adjunct of the identity morphism is the adjunction unit, as shown.

The second equality is fomally dual:

Hom 𝒟(CX,CX) ()˜ Hom 𝒞(LCX,X) Hom 𝒟(CX,C(f)) Hom X(LCX,f) Hom 𝒟(CX,CY) ()˜ Hom 𝒞(LCX,Y)AAAAA{CXid CXCX} {LCXϵ X X} {CXC(f)C(Y)} {LCXfϵ X Y} \array{ Hom_{\mathcal{D}}( C X, C X) &\overset{\widetilde { (-) }}{\longrightarrow}& Hom_{\mathcal{C}}( L C X , X) \\ {}^{\mathllap{ Hom_{\mathcal{D}}( C X, C(f) ) }} \big\downarrow && \big\downarrow^{ \mathrlap{ Hom_{X}( L C X, f ) } } \\ Hom_{\mathcal{D}}( C X, C Y ) &\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\} }

\,

\,

We now consider a sequence of equivalent reformulations of the condition of adjointness.

Proposition

(general adjuncts in terms of unit/counit)

Consider a pair of adjoint functors

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

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=η cR(f˜)AAAA c η c f R(L(c)) R(f˜) R(d) L(c) f˜ d f \;=\; \eta_c \circ R(\widetilde f) \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 0Grp Set \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{\pi_0}{\longleftarrow}} {\bot} Grp

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 XιL. X \overset{\eta_X}{\longrightarrow} \iota \circ L \,.

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

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

(CR)(CL) \left( \Box \;\coloneqq\; C \circ R \right) \;\dashv\; \left( \bigcirc \;\coloneqq\; C \circ L \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 )AAAAAAAΓX 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{ \Gamma 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. :

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

We say:

  1. This provides right Aufhebung of the opposition exhibited by box 1 1\box_1 \dashv \bigcirc_1 if there is also the diagonal inclusion

    box 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 box 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 id 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{id}}\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