nLab locally presentable (infinity,1)-category

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

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

See also at locally presentable categories - introduction.

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

Terminological variant. The term “$\kappa$-compactly generated (∞,1)-category” is sometimes used to mean “locally $\kappa$-presentable (∞,1)-category. See there for a discussion of usage differences.

Definition

Definition

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

1. it is accessible

2. it has all small (∞,1)-colimits.

Proposition

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

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

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

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

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

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

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

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

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

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

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

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

This is Lurie, Thm. 5.5.1.1 & Prop. A.3.7.6, following Simpson 99, Dugger 00; review in Cisinski 19, Thm. 7.11.16, Rem. 7.11.17.

Remark

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

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

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

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

This is Lurie, def. 5.5.3.1.

This $(\infty,1)$-category $Pr(\infty,1)Cat$ 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. .

Lemma

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

$$Ind_\kappa(f) \colon Ind_\kappa(\mathcal{C}) \to Ind_\kappa(\mathcal{D})$$

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

This is (Lurie, lemma 5.5.1.3).

Lemma

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

This is Lurie, lemma 5.5.1.4.

(…)

Stability under various constructions

Proposition

For $C$ a locally presentable $(\infty,1)$-category and $p : K \to C$ a diagram in $C$, also the over (∞,1)-category $C_{/p}$ as well as the under-$(\infty,1)$-category $C_{p/}$ 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 $(\infty,1)$-categories by forming limits over given ones. In particular the product of locally presentable $(\infty,1)$-categories is again locally presentable.

Limits and colimits

In the first definition of locally presentable $(\infty,1)$-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 $C^{op} \to \infty Grpd$ preserves limits (see (∞,1)-Yoneda embedding). If $C$ is locally presentable, then also the converse holds:

Proposition

If $\mathcal{C}$ is a locally presentable $(\infty,1)$-category then an (∞,1)-functor $C^{op} \to \infty Grpd$ 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 $F : C^{op} \to \infty Grpd$ is representable. By the above characterizations we know that $C$ is an accessible localization of a presheaf category.

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

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

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

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

the functor represented by $f$.

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

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

$$F \circ L^{op} : PSh(D)^{op} \stackrel{L^{op}}{\to} C^{op} \stackrel{F}{\to} \infty Grpd$$

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

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

This statement has the following important consequence:

Corollary

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

This is HTT, prop. 5.5.2.4.

Proof

We may compute the limit after applying the (∞,1)-Yoneda embedding $j : C \to PSh_{(\infty,1)}(c)$. Since this is a full and faithful (∞,1)-functor it is sufficient to check that the limit computed in $PSh(C)$ lands in the essential image of $j$. 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 $I \to PSh(C)$ a diagram of limit-preserving functors that $\lim_i F_i$ is a functor that commutes with all limits, let $a : J \to C$ be a diagram and compute (verbatim as in ordinary category theory)

$$\begin{aligned} \lim_j (\lim_i F_i)(a_j) & \simeq \lim_j (\lim_i F_i(a_j)) \\ & \simeq \lim_i (\lim_j F_i(a_j)) \\ & \simeq \lim_i F_i(\lim a_j) \\ & \simeq (\lim_i F_i)(\lim a_j) \end{aligned} \,.$$

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

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

(See also at Ho(CombModCat)).

Remark

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

$C^\circ$ and $D^\circ$ are equivalent as (∞,1)-categories precisely if they are connected by a zig-zag of simplicial Quillen equivalences

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

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 $(\infty,1)$-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 $S$ by the simplicial model category of simplicial presheaves on $S$.

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 $(\infty,1)$-category where the localization functor also preserves finite limits.

Proposition

For $C$ and $D$ locally presentable $(\infty,1)$-categories, write $Func^L(C,D) \subset Func(C,D)$ for the full sub-$(\infty,1)$-category on left-adjoint $(\infty,1)$-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 $C$ an $(\infty,1)$-category with finite products, the $(\infty,1)$-category $Alg_{(\infty,1)}(C)$ of algebras over $C$ regarded as an (∞,1)-algebraic theory is locally presentable.

Related concepts

• Ho(CombModCat)

Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible reflective localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.

$\phantom{A}$(n,r)-categories$\phantom{A}$	$\phantom{A}$toposes$\phantom{A}$	locally presentable	loc finitely pres	localization theorem	free cocompletion	accessible
(0,1)-category theory	locales	suplattice	algebraic lattices	Porst’s theorem	powerset	poset
category theory	toposes	locally presentable categories	locally finitely presentable categories	Gabriel–Ulmer’s theorem	presheaf category	accessible categories
model category theory	model toposes	combinatorial model categories		Dugger's theorem	global model structures on simplicial presheaves	n/a
(∞,1)-category theory	(∞,1)-toposes	locally presentable (∞,1)-categories		Simpson’s theorem	(∞,1)-presheaf (∞,1)-categories	accessible (∞,1)-categories

• compactly generated (infinity,1)-category

References

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

• Carlos Simpson, A Giraud-type characterization of the simplicial categories associated to closed model categories as $\infty$-pretopoi (arXiv:math/9903167)

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

• Jacob Lurie, Section 5 of: Higher Topos Theory

with section A.3.7 establishing the relation to 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 $(\infty,1)$-categories as localizations of $(\infty,1)$-presheaf categories is a variant – is due to

• Dan Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001), no. 1, 177-201 (arXiv:math/0007068)

Further textbook account:

• Denis-Charles Cisinski, Section 7.11 in: Higher category theory and homotopical algebra, Cambridge University Press 2019 (doi:10.1017/9781108588737, pdf)

Exposition and review:

• topology@ub, Presentable infinity categories, seminar notes, 2019

On factorication of functors between presentable $\infty$-categories as coreflections followed by a tower of monadic functors:

• Lior Yanovski, The Monadic Tower for $\infty$-Categories (arXiv:2104.01816)

Discussion internal to any (∞,1)-topos:

• Louis Martini, Sebastian Wolf, Presentable categories internal to an ∞-topos [arXiv:2209.05103]