nLab locally presentable categories - introduction

This page means to give an introduction to the notion of locally presentable category, and its related notions in higher category theory and survey some fundamental properties.

Expected background of the reader:

• For the first section Basic idea in category theory the reader is assumed to be familiar with basic notions of category theory such as presheaves and colimits.

• For the section Basic idea in model category theory the reader is assumed to be familiar with basic notions in model category theory such as cofibrantly generated model categories and homotopy colimits.

• For the third section Basic idea in (∞,1)-category theory the reader is assumed to be familiar with basic concepts of (∞,1)-category theory such as (∞,1)-categories of (∞,1)-presheaves and (∞,1)-colimits.

Basic idea in category theory

The general idea is that a locally presentable category is a large category generated from small data: from small objects under small colimits.

Generation from generators

The notion of locally presentable category is, at least roughly, an analogue for categories of the notion of a finitely generated module.

Example

An abelian group $A$ is called finitely generated if there is a finite subset

$$\iota \colon S \hookrightarrow U(A)$$

of the underlying set $U(A)$ of $A$, such that every element of $A$ is a sum of such generating elements.

Remark

We always have the maximal such presentation where $S = U(A)$ is the whole underlying set and $\iota \colon F(U(A)) \to A$ is the counit of the free-forgetful adjunction. But the presentation is all the more interesting/useful the smaller $S$ is.

Now, the categorification of “commutative sum” is colimit. Hence let now $\mathcal{C}$ be a category with all small colimits.

Definition

We say a subclass $S \hookrightarrow Obj(\mathcal{C})$ of objects or equivalently the full subcategory $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on this subclass generates $\mathcal{C}$ if every object in $\mathcal{C}$ is a colimit of objects in $\mathcal{C}^0$, hence the colimit over a diagram of the form

$$D \to \mathcal{C}^0 \hookrightarrow \mathcal{C} \,.$$

As before, such a presentation is all the more useful the “smaller” the generating data is. In order to grasp the various aspects of the notion of “smallness” in category theory we need to recall the notion of regular cardinal.

Small data

Definition

The cardinality $\kappa = {\vert S\vert}$ of a set $S$ is regular if every coproduct/disjoint union of sets of cardinality smaller than $\kappa$ and indexed by a set of cardinality smaller than $\kappa$ is itself of cardinality smaller than $\kappa$.

Example

The smallest regular cardinal is $\aleph_0 = {\vert \mathbb{N}\vert}$: every finite union of finite sets is itself a finite set. (See the entry on regular cardinals for a discussion as to whether one might consider some finite cardinals as being `regular'.)

We can now speak of objects that are “$\kappa$-small sums” using the notion of $\kappa$-filtered colimits:

Definition

For $\kappa$ a regular cardinal, a $\kappa$-filtered category is one where every diagram of size $\lt \kappa$ has a cocone.

Example

In an $\aleph_0$-filtered category every finite diagram has a cocone. This is equivalent to:

1. for every pair of objects there is a third object such that both have a morphism to it;

2. for every pair of parallel morphisms there is a morphism out of their codomain such that the two composites are equal.

Example

The tower diagram category $(\mathbb{N}, \leq)$

$$X_0 \to X_1 \to X_2 \to \cdots$$

is filtered.

Remark

For $\lambda \gt \kappa$ a bigger regular cardinal, every $\lambda$-filtered category is in particular also $\kappa$-filtered.

Using this we have the central definition now:

Definition

A $\kappa$-filtered colimit is a colimit over a $\kappa$-filtered diagram.

A crucial characterizing property of $\kappa$-filtered colimits is the following:

Proposition

A colimit in Set is $\kappa$-filtered precisely if it commutes with all $\kappa$-small limits.

In particular a colimit in Set is filtered (meaning: $\aleph_0$-filtered) precisely if it commutes with all finite limits.

Definition

An object $A \in \mathcal{C}$ is a $\kappa$-compact object if it commutes with $\kappa$-filtered colimits, hence if for $X \colon I \to \mathcal{C}$ any $\kappa$-filtered diagram, the canonical function

$$\underset{\to_i}{\lim} \mathcal{C}(A,X_i) \to \mathcal{C}(A, \underset{\to_i}{\lim} X_i)$$

is a bijection.

We say $X$ is a small object if it is $\kappa$-compact for some regular cardinal $\kappa$.

Remark

If $\lambda \gt \kappa$, then being $\lambda$-compact is a weaker condition than being $\kappa$-compact.

Remark

The object $A$ commutes with the colimit over $I$ precisely if every morphism $A \to \underset{\to_i}{\lim} X_i$ lifts to a morphism $A \to X_j$ into one of the $X_j$. Schematically, depicting specifically a sequential colimit, this means that we have:

$$\array{ \cdots&\to&X_{j-1} &\to& X_j &\to& X_{j+1} &\to& \cdots \\ &&&{}^{\mathllap{\exists \hat f}}\nearrow&\downarrow & \swarrow \\ &&A& \stackrel{f}{\to} & \underset{\to}{\lim} X_i } \,.$$

Hence $A$ is “small enough” such that mapping it into the sum of all the $X_i$ it always entirely lands inside one of the $X_i$.

Remark

There is a close relation between the notion of “compact” as in, on the one hand, compact topological space and compact topos, and on the other as in compact object as above. This is mediated by proposition . But the relation is a bit more subtle and takes a bit more discussion than we maybe want to get into here.

Locally presentable category: generated from colimits over small objects

Using this we can now say:

Definition

A locally small category $\mathcal{C}$ is a locally presentable category if it has all small colimits and there is a small set $S \hookrightarrow Obj(\mathcal{C})$ of small objects such that this generates $\mathcal{C}$, by def. .

Remark

The adjective “locally” in “locally presentable category” is to indicate that the condition is all about the objects, only. There is a different notion of “presented category”.

There are a bunch of equivalent reformulations of the notion of locally presentable category. One of the more important ones we again motivate first by analogy with presentable modules:

Generation exhibited by epimorphism from a free object

Example

If an abelian group $A$ is generated by a set $S \hookrightarrow U(A)$ as in example , this means equivalently that there is an epimorphism

$$L \colon F(S) \to A$$

from the free abelian group $F(S)$ generated by $S$, hence the group obtained by forming formal sums of elements in $S$. Here the epimorphism sends formal sums to actual sums in $A$:

$$L(\sum_k s_k) \coloneqq \sum_k \iota(s_k) \,.$$

Remark

The categorification of the notion free abelian group is the notion of free cocompletion of a category $\mathcal{C}^0$: the category of presheaves $PSh(\mathcal{C}^0)$.

Accordingly:

Example

If a full subcategory $\iota \colon \mathcal{C}^0 \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under colimits as in defn. , then there is a functor

$$L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$$

which sends formal colimits to actual colimits in $\mathcal{C}$

$$L(\underset{\to_k}{\lim} s_k) \coloneqq \underset{\to_k}{\lim} \iota(s_k) \,.$$

Here $L$ by construction preserves all colimits.

Therefore conversely, given a colimit-preserving functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ we want to say that it locally presents $\mathcal{C}$ if $L$ is “suitably epi”.

It turns out that “suitably epi” is to be the following:

Definition

A functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from the category of presheaves over a small category $\mathcal{C}^0$ is an accessible localization if

• $L$ has a section, hence a functor $R \colon \mathcal{C} \to PSh(\mathcal{C}^0)$ with a natural isomorphism $L\circ R \simeq id_{\mathcal{C}}$;

• such that

  1. $R$ is right adjoint to $L$;

  2. $R\circ L$ preserves $\kappa$-filtered colimits.

With this notion we have the following analog of the familiar statement that an abelian group is generated by $S$ precisely if there is an epimorphism $L \colon F(S) \to A$:

Theorem

A category $\mathcal{C}$ is locally presentable according to def. precisely if it is an accessible localization, def. ,

$$L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$$

for some small category $\mathcal{C}^0$.

This is due to (Adámek-Rosický, prop 1.46).

Left exact localizations

Remark

A locally presentable category $\mathcal{C}$ is called a topos, precisely if the localization functor $L \colon PSh(\mathcal{C}^0) \to \mathcal{C}$ from theorem in addition is a left exact functor, meaning that it preserves finite limits.

Summary and overview

In summary the discussion above says that the notion of locally presentable categories sits in a sequence of notions as indicated in the row labeled “category theory” in the following table. The other rows are supposed to indicate that regarding a category as a (1,1)-category and simply varying in this story the parameters $(n,r)$ in “(n,r)-category” one obtains fairly straightforward analogs of the notion of locally presentable category in other fragments of higher category theory. These we discuss in more detail further below.

Locally presentable categories: Large categories whose objects arise from small generators under small relations.

(n,r)-categories …	satisfying Giraud's axioms	inclusion of left exact localizations	generated under colimits from small objects		localization of free cocompletion		generated under filtered colimits from small objects
(0,1)-category theory	(0,1)-toposes	$\hookrightarrow$	algebraic lattices	$\simeq$ Porst’s theorem	subobject lattices in accessible reflective subcategories of presheaf categories
category theory	toposes	$\hookrightarrow$	locally presentable categories	$\simeq$ Adámek-Rosický‘s theorem	accessible reflective subcategories of presheaf categories	$\hookrightarrow$	accessible categories
model category theory	model toposes	$\hookrightarrow$	combinatorial model categories	$\simeq$ Dugger’s theorem	left Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory	(∞,1)-toposes	$\hookrightarrow$	locally presentable (∞,1)-categories	$\simeq$ Simpson’s theorem	accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories	$\hookrightarrow$	accessible (∞,1)-categories

Basic idea in model category theory

Model structure on simplicial presheaves

The analog of a category of presheaves in model category theory is the model structure on simplicial presheaves, which we now briefly indicate.

Write sSet for the category of simplicial sets. Here we always regard this as equipped with the standard model structure on simplicial sets $sSet_{Quillen}$.

Definition

For $C$ a small category write $[C^{op}, sSet]\simeq [C^{op}, Set]^{\Delta^{op}}$ for the category of simplicial presheaves. The global projective model structure on simplicial presheaves $[C^{op}, sSet]_{proj}$ has as

• weak equivalences the objectwise weak homotopy equivalences of simplicial sets

• fibrations the objectwise Kan fibrations.

Remark

Accordingly $[C^{op}, sSet]$ is a cofibrantly generated model category with generating (acyclic) cofibrations the tensoring of objects of $C$ with the generating (acyclic) cofibrations of $sSet_{Quillen}$.

Left Bousfield localization

Definition

Given a model category $[C^{op}, Set]_{proj}$ and set $\mathcal{S} \subset Mor([C^{op}, Set])$ of morphisms, the left Bousfield localization is the model structure with the same cofibrations and weak equivalences the $\mathcal{S}$-local morphisms.

$$[C^{op}, Set]_{proj,\mathcal{S}} \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} [C^{op}, Set]_{proj} \,.$$

Combinatorial model categories

The simple idea of the following definition is to say that the model category analog of locally presentable category is simply a model structure on a locally presentable category.

Definition

A model category is a combinatorial model category if

1. the underlying category is a locally presentable category;

2. the model structure is a cofibrantly generated model category.

Dugger’s theorem

Theorem

Every combinatorial model category, def. , is Quillen equivalent to a left Bousfield localization of a global model structure on simplicial presheaves as in def. .

See at combinatorial model category - Dugger’s theorem.

Basic idea in $(\infty,1)$-category theory

$(\infty,1)$-Presheaves

Definition

For $\mathcal{C}$ and $\mathcal{D}$ two (∞,1)-categories and $\mathcal{C}_{s}, \mathcal{D}_s \in sSet$ two models as quasi-categories, an (∞,1)-functor $F \colon \mathcal{C} \to \mathcal{D}$ is simply a homomorphism of simplicial set $\mathcal{C}_s \to \mathcal{D}_s$.

The (∞,1)-category of (∞,1)-functors $Func(\mathcal{C}, \mathcal{D})_s$ as a quasi-category is simply the hom object of simplicial set

$$Func(\mathcal{C}, \mathcal{D})_s = sSet(\mathcal{C}_s, \mathcal{D}_s) \in QuasiCat \hookrightarrow sSet \,.$$

Definition

For $\mathcal{D}$ an (∞,1)-category, the (∞,1)-category of (∞,1)-presheaves on $\mathcal{D}$ is the functor category

$$PSh_\infty(\mathcal{D}) = Func(\mathcal{D}^{op}, \infty Grpd)$$

out of the opposite (∞,1)-category of $\mathcal{D}$ into the (∞,1)-category of ∞-groupoids.

Localizations of $(\infty,1)$-categories

The notions of adjoint functors, full and faithful functors etc. have straightforward, essentially verbatim generalizations to $(\infty,1)$-categories:

Definition

A pair of (∞,1)-functors

$$C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$$

is a pair of adjoint (∞,1)-functors, if there exists a unit transformation $\epsilon : Id_C \to R \circ L$ – a morphism in the (∞,1)-category of (∞,1)-functors $Func(C,D)$ – such that for all $c \in C$ and $d \in D$ the induced morphism

$$Hom_C(L(c),d) \stackrel{R_{L(c), d}}{\to} Hom_D(R(L(c)), R(d)) \stackrel{Hom_D(\epsilon, R(d))}{\to} Hom_D(c,R(d))$$

is an equivalence of ∞-groupoids.

Definition

An (∞,1)-functor $F \colon \mathcal{C} \to \mathcal{D}$ is a full and faithful (∞,1)-functor if for all objects $X,Y \in \mathcal{C}$ the component

$$F_{X,Y} \colon \mathcal{C}(X,Y) \stackrel{\simeq}{\to} \mathcal{D}(F(X), F(Y))$$

is an equivalence of ∞-groupoids.

Definition

A reflective sub-(∞,1)-category $\mathcal{C} \hookrightarrow \mathcal{D}$ is a full and faithful (∞,1)-functor with a left adjoint (∞,1)-functor.

Locally presentable $(\infty,1)$-categories

We have then the essentially verbatim analog of the situation for ordinary categories:

Definition

An (∞,1)-category $\mathcal{C}$ is a locally presentable (∞,1)-category if there exists a small set of objects such that the full sub-(∞,1)-category $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ on it generates $\mathcal{C}$ under (∞,1)-colimits.

And the equivalent characterization is now as before

Theorem

An (∞,1)-category is a locally presentable (∞,1)-category, def. , precisely if it is equivalent to localization, def. ,

$$\mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} PSh_\infty(\mathcal{C}^0)$$

of an (∞,1)-category of (∞,1)-presheaves, def. , such that $R \circ L$ preserves $\kappa$-filtered (∞,1)-colimits for some regular cardinal $\kappa$.

This appears as Lurie, theorem 5.5.1.1, attributed there to Carlos Simpson.

$(\infty,1)$-Toposes

As before, if a locally presentable $(\infty,1)$-category arises as the localization $L \colon PSh_\infty(\mathcal{C}^0) \to \mathcal{C}$ of a left exact (∞,1)-functor, then it is an (∞,1)-topos.

Presentation by combinatorial model categories

There is a close match between the theory of combinatorial model categories and locally presentable (∞,1)-categories.

Theorem

Every locally presentable (∞,1)-category arises, up to equivalence of (∞,1)-categories, as the simplicial localization of a combinatorial model category.

This is part of Lurie, theorem 5.5.1.1.

Accordingly, every simplicial Quillen adjunction between combinatorial model categories gives rise to a pair of adjoint (∞,1)-functors between the corresponding locally presentable $(\infty,1)$-categories.

Hence a left Bousfield localization of a model structure on simplicial presheaves presents a corresponding localization of an (∞,1)-category of (∞,1)-presheaves to a locally presentable (∞,1)-category.

$$\array{ \mathcal{C} &\stackrel{\overset{}{\leftarrow}}{\hookrightarrow}& PSh_\infty(C) \\ \left[C^{op}, sSet\right]_{proj,\mathcal{S}} &\stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}}& [C^{op}, sSet]_{proj} }$$

References

The standard textbook for locally presentable categories is

• Jiří Adámek, Jiří Rosický, Locally presentable and accessible categories, Cambridge University Press, (1994)

Decent accounts of combinatorial model categories include secton A.2.6 of

• Jacob Lurie, Higher Topos Theory

and

• Clark Barwick, On (Enriched) Left Bousfield Localization of Model Categories (arXiv:0708.2067)

The standard text for locally presentable (∞,1)-categories is section 5 of Lurie.