nLab
locally presentable (infinity,1)-category

Contents

Idea

An (∞,1)-category is called locally presentable if it has all small (∞,1)-colimits and its objects are presented under (∞,1)-colimits by a small set of small objects. This is the direct analog in (∞,1)-category theory of the notion of locally presentable category in category theory.

There is a wealth of equivalent ways to make precise what this means, which are listed below. Two particularly useful ones are:

  1. A locally presentable (,1)(\infty,1)-category is an accessible (∞,1)-category that admits all small (∞,1)-colimits.

  2. The locally presentable (,1)(\infty,1)-categories 𝒞\mathcal{C} are precisely the accessibly embedded localizations/reflections 𝒞PSh (K)\mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K) of an (∞,1)-category of (∞,1)-presheaves. In particular, if the reflector of this reflection is a left exact (∞,1)-functor, then 𝒞\mathcal{C} is an (∞,1)-topos.

See also at locally presentable categories - introduction.

Warning on terminology. In Lurie the term presentable (,1)(\infty,1)-category is used for what we call a locally presentable (,1)(\infty,1)-category here, in order to be in line with the established terminology of locally presentable category in ordinary category theory.

Definition

Definition

An (∞,1)-category 𝒞\mathcal{C} is called locally presentable if

  1. it is accessible

  2. it has all small (∞,1)-colimits.

Proposition

That 𝒞\mathcal{C} is locally presentable is equivalent to each of the following equivalent characterizations.

  1. 𝒞\mathcal{C} is locally small, with all small (∞,1)-colimits such that there is a small set SObj(𝒞)S \hookrightarrow Obj(\mathcal{C}) of small objects which generates all of 𝒞\mathcal{C} under (∞,1)-colimits.

  2. 𝒞\mathcal{C} is the localization of an (∞,1)-category of (∞,1)-presheaves PSh (K)PSh_\infty(K) along an accessible (∞,1)-functor:

    there exists a small (∞,1)-category KK and a pair of adjoint (∞,1)-functors

    𝒞PSh (K) \mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)

    such that the right adjoint 𝒞PSh (K)\mathcal{C} \hookrightarrow PSh_\infty(K) is full and faithful and accessible.

    (if here in addition ff is left exact then 𝒞\mathcal{C} is an (∞,1)-category of (∞,1)-sheaves on KK).

  3. There exists a combinatorial simplicial model category AA and and equivalence of (infinity,1)-categories 𝒞L WA\mathcal{C} \simeq L_W A with the simplicial localization of AA.

    More explicitly: with 𝒞\mathcal{C} incarnated as a quasi-category there is equivalence of quasi-categories 𝒞N(A ) \mathcal{C} \simeq N(A^\circ) of 𝒞\mathcal{C} with the homotopy coherent nerve of the full sSet-enriched subcategory of AA on fibrant and cofibrant objects.

  4. 𝒞\mathcal{C} is accessible and for every regular cardinal κ\kappa the full sub-(∞,1)-category 𝒞 κ𝒞\mathcal{C}^\kappa \hookrightarrow \mathcal{C} on the κ\kappa compact objects admits κ\kappa-small (∞,1)-colimits.

  5. There exists a regular cardinal κ\kappa such that 𝒞\mathcal{C} is κ\kappa-accessible and C κC^\kappa admits κ\kappa-small colimits;

  6. There exists a regular cardinal κ\kappa, a small (,1)(\infty,1)-category DD with κ\kappa-small colimits and an equivalence Ind κD𝒞Ind_\kappa D \stackrel{\simeq}{\to} \mathcal{C} with the category of κ\kappa-ind-objects of DD.

This is Lurie, theorem 5.5.1.1, following (Simpson). We discuss this further below in Equivalent characterizations.

Remark

That localizations 𝒞PSh (,1)(K)\mathcal{C} \stackrel{\leftarrow}{\hookrightarrow} PSh_{(\infty,1)}(K) correspond to combinatorial simplicial model categories is essentially Dugger’s theorem (Dugger): every combinatorial model category arises, up to Quillen equivalence, as the left left Bousfield localization of the global projective model structure on simplicial presheaves.

Locally presentable (,1)(\infty,1)-categories have a number of nice properties, and therefore it is of interest to consider as morphisms between them only those (∞,1)-functors that preserve these properties. It turns out that it is useful to consider colimit preserving functors. By the adjoint (∞,1)-functor theorem these are precisely the functors that have a right adjoint (∞,1)-functor.

Definition

Write Pr(∞,1)Cat \subset (∞,1)Cat for the (non-full) sub-(∞,1)-category of (∞,1)Cat (the collection of not-necessarily small (,1)(\infty,1)-categories) on

  • those objects that are locally presentable (,1)(\infty,1)-categories;

  • those morphisms that are colimit-preserving (∞,1)-functors.

This is Lurie, def. 5.5.3.1.

This (,1)(\infty,1)-category Pr(,1)CatPr(\infty,1)Cat in turn as special properties. More on that is at symmetric monoidal (∞,1)-category of presentable (∞,1)-categories.

Properties

Equivalent characterizations

We indicate stepts in the proof of prop. 1.

Lemma

Let f:𝒞𝒟f \colon \mathcal{C} \to \mathcal{D} be an (∞,1)-functor which exhibits 𝒟\mathcal{D} as an idempotent completion of? 𝒞\mathcal{C}. Let κ\kappa be a regular cardinal. Then the induced functor on (∞,1)-categories of ind-objects

Ind κ(f):Ind κ(𝒞)Ind κ(𝒟) Ind_\kappa(f) \colon Ind_\kappa(\mathcal{C}) \to Ind_\kappa(\mathcal{D})

is an equivalence of (∞,1)-categories.

This is (Lurie, lemma 5.5.1.3).

Lemma

Let L:𝒞𝒟L \colon \mathcal{C} \to \mathcal{D} be an (∞,1)-functor between (∞,1)-categories which have κ\kappa-filtered (∞,1)-colimits, and let RR be a right adjoint (∞,1)-functor of LL. If RR preserves κ\kappa-filtered (∞,1)-colimits then LL preserves κ\kappa-compact objects.

This is Lurie, lemma 5.5.1.4.

(…)

Stability under various constructions

Proposition

For CC a locally presentable (,1)(\infty,1)-category and p:KCp : K \to C a diagram in CC, also the over (∞,1)-category C /ppC_{/pp} as well as the under-(,1)(\infty,1)-category C p/C_{p/} are locally presentable.

This is HTT, prop. 5.5.3.10, prop. 5.5.3.11.

Example

Since Pr(∞,1)Cat admits all small limits, we obtain new locally presentable (,1)(\infty,1)-categories by forming limits over given ones. In particular the product of locally presentable (,1)(\infty,1)-categories is again locally presentable.

Limits and colimits

In the first definition of locally presentable (,1)(\infty,1)-category above only the existence of colimits is postulated. An important fact is that it follows automatically that also all small limits exist:

A representable functor C opGrpdC^{op} \to \infty Grpd preserves limits (see (∞,1)-Yoneda embedding). If CC is locally presentable, then also the converse holds:

Proposition

If 𝒞\mathcal{C} is a locally presentable (,1)(\infty,1)-category then an (∞,1)-functor C opGrpdC^{op} \to \infty Grpd is a representable functor precisely if it preserves limits.

This is HTT, prop. 5.5.2.2.

Proof

We need to prove that a limit-preserving functor F:C opGrpdF : C^{op} \to \infty Grpd is representable. By the above characterizations we know that CC is an accessible localization of a presheaf category.

So consider first the case that C=PSh(D)C = PSh(D) is a presheaf category. Write

f:D opj opPSh(D) opFGrpd f : D^{op} \stackrel{j^{op}}{\to} PSh(D)^{op} \stackrel{F}{\to} \infty Grpd

for the precomposition of FF with the (∞,1)-Yoneda embedding. Then let

F:=Hom C(,f):PSh(D) opGrpd F' := Hom_{C}(-,f) : PSh(D)^{op} \to \infty Grpd

the functor represented by ff.

We claim that FFF \simeq F', which proves that FF is represented by Fj opF \circ j^{op}: since both FF and FF' preserve limits (hence colimits as functors on PSh(D)PSh(D)) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of DD that it is sufficient to show that Fj opFj opF \circ j^{op} \simeq F' \circ j^{op}. But this is the case precisely by the statement of the full (∞,1)-Yoneda lemma.

Now consider more generally the case that CC is a reflective sub-(∞,1)-category of PSh(D)PSh(D). Let L:PSh(D)CL : PSh(D) \to C be the left adjoint reflector. Since it respects all colimits, the composite

FL op:PSh(D) opL opC opFGrpd F \circ L^{op} : PSh(D)^{op} \stackrel{L^{op}}{\to} C^{op} \stackrel{F}{\to} \infty Grpd

respects all limits. By the above it is therefore represented by some object XPSh(D)X \in PSh(D).

By the general properties of reflective sub-(∞,1)-categories, we have that CC is the full sub-(∞,1)-category of PSh(D)PSh(D) on those objects that are local objects with respect to the morphisms that LL sends to equivalences. But XX, since it presents FL opF \circ L^{op}, is manifestly local in this sense and therefore also represents FL op| CF \circ L^{op}|_{C}. But on CC the functor LL is equivalent to the identity, so that this is equivlent to FF.

This statement has the following important consequence:

Corollary

A locally presentable (,1)(\infty,1)-category CC has all small limits.

This is HTT, prop. 5.5.2.4.

Proof

We may compute the limit after applying the (∞,1)-Yoneda embedding j:CPSh (,1)(c)j : C \to PSh_{(\infty,1)}(c). Since this is a full and faithful (∞,1)-functor it is sufficient to check that the limit computed in PSh(C)PSh(C) lands in the essential image of jj. But by the above lemma, this amounts to checking that the limit over limit-preserving functors is itself a limit-preserving functor. This follows using that limits of functors are computed objectwise and that generally limits commute with each other (see limit in a quasi-category):

to check for IPSh(C)I \to PSh(C) a diagram of limit-preserving functors that lim iF i\lim_i F_i is a functor that commutes with all limits, let a:JCa : J \to C be a diagram and compute (verbatim as in ordinary category theory)

lim j(lim iF i)(a j) lim j(lim iF i(a j)) lim i(lim jF i(a j)) lim iF i(lima j) (lim iF i)(lima j). \begin{aligned} \lim_j (\lim_i F_i)(a_j) & \simeq \lim_j (\lim_i F_i(a_j)) \\ & \simeq \lim_i (\lim_j F_i(a_j)) \\ & \simeq \lim_i F_i(\lim a_j) \\ & \simeq (\lim_i F_i)(\lim a_j) \end{aligned} \,.

As (,1)(\infty,1)-categories presented by combinatorial simplicial model categories

By prop. 1 locally presentable (,1)(\infty,1)-categories are equivalently those (∞,1)-categories which are presented by a combinatorial simplicial model category CC in that they are the full simplicial subcategory C CC^\circ \hookrightarrow C on fibrant-cofibrant objects of CC (or, equivalently, the quasi-category associated to this simplicially enriched category).

Remark

Under this presentation, equivalence of (∞,1)-categories between locally presentable (,1)(\infty,1)-categories corresponds to zigzags of Quillen equivalences between presenting combinatorial simplicial model categories:

C C^\circ and D D^\circ are equivalent as (,1)(\infty,1)-categories precisely if there exists a chain of simplicial Quillen equivalence

CD. C \stackrel{\leftarrow}{\to} \stackrel{\to}{\leftarrow} \stackrel{\leftarrow}{\to} \cdots D.

This is Lurie, remark A.3.7.7.

Remark

Partly due to the fact that simplicial model categories have been studied for a longer time – partly because they are simply more tractable than (∞,1)-categories – many (,1)(\infty,1)-categories are indeed handled in terms of such a presentation by a simplicial model category.

The canonical example is the presentation of the (∞,1)-category of (∞,1)-sheaves on an ordinary (1-categorical) site SS by the simplicial model category of simplicial presheaves on SS.

Examples

The basic example is:

Example

∞Grpd is locally presentable.

(Lurie, example 5.5.1.8)

Proof

According to the discussion at (∞,1)-colimit – Tensoring with an ∞-groupoid every ∞-groupoid is the colimit over itself of the functor contant on the point, the terminal \infty-groupoid. This is clearly compact, and hence generates ∞Grpd.

Example

An (∞,1)-topos is precisely a locally presentable (,1)(\infty,1)-category where the localization functor also preserves finite limits.

Proposition

For CC and DD locally presentable (,1)(\infty,1)-categories, write Func L(C,D)Func(C,D)Func^L(C,D) \subset Func(C,D) for the full sub-(,1)(\infty,1)-category on left-adjoint (,1)(\infty,1)-functors. This is itself locally presentable

This is HTT, prop 5.5.3.8

Notice that this makes the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories closed .

Proposition

For CC an (,1)(\infty,1)-category with finite products, the (,1)(\infty,1)-category Alg (,1)(C)Alg_{(\infty,1)}(C) of algebras over CC regarded as an (∞,1)-algebraic theory is locally presentable.

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

(n,r)-categoriessatisfying Giraud's axiomsinclusion of left exaxt localizationsgenerated under colimits from small objectslocalization of free cocompletiongenerated under filtered colimits from small objects
(0,1)-category theory(0,1)-toposes\hookrightarrowalgebraic lattices\simeq Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes\hookrightarrowlocally presentable categories\simeq Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories\hookrightarrowaccessible categories
model category theorymodel toposes\hookrightarrowcombinatorial model categories\simeq Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes\hookrightarrowlocally presentable (∞,1)-categories\simeq
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories\hookrightarrowaccessible (∞,1)-categories

References

The theory of locally presentable (,1)(\infty,1)-categories was first implicitly conceived in terms of model category presentations in

The full intrinsic (,1)(\infty,1)-categorical theory appears in section 5

with section A.3.7 establishing the relation combinatorial model categories and Dugger’s theorem in HTT, prop A.3.7.6

The statement of Dugger’s theorem of which the characterization of locally presentable (,1)(\infty,1)-categories as localizations of (,1)(\infty,1)-presheaf categories is a variant is due to

Revised on February 10, 2014 06:50:19 by Urs Schreiber (89.204.135.153)