# nLab locally presentable (infinity,1)-category

### Context

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

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

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

Warning on terminology. In Lurie the term presentable $\left(\infty ,1\right)$-category is used for what we call a locally presentable $\left(\infty ,1\right)$-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 $𝒞$ is called locally presentable if

1. it is accessible

2. it has all small (∞,1)-colimits.

###### Proposition

That $𝒞$ is locally presentable is equivalent to each of the following equivalent characterizations.

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

2. $𝒞$ is the localization of an (∞,1)-category of (∞,1)-presheaves ${\mathrm{PSh}}_{\infty }\left(K\right)$ along an accessible (∞,1)-functor:

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

$𝒞\stackrel{\stackrel{}{←}}{↪}{\mathrm{PSh}}_{\infty }\left(K\right)$\mathcal{C} \stackrel{\overset{}{\leftarrow}}{\hookrightarrow} PSh_\infty(K)

such that the right adjoint $𝒞↪{\mathrm{PSh}}_{\infty }\left(K\right)$ is full and faithful and accessible.

(if here in addition $f$ is left exact then $𝒞$ 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 $𝒞\simeq {L}_{W}A$ with the simplicial localization of $A$.

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

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

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

6. There exists a regular cardinal $\kappa$, a small $\left(\infty ,1\right)$-category $D$ with $\kappa$-small colimits and an equivalence ${\mathrm{Ind}}_{\kappa }D\stackrel{\simeq }{\to }𝒞$ with the category of $\kappa$-ind-objects of $D$.

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

###### Remark

That localizations $𝒞\stackrel{←}{↪}{\mathrm{PSh}}_{\left(\infty ,1\right)}\left(K\right)$ 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 $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-categories) on

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

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

This is Lurie, def. 5.5.3.1.

This $\left(\infty ,1\right)$-category $\mathrm{Pr}\left(\infty ,1\right)\mathrm{Cat}$ 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 $f:𝒞\to 𝒟$ be an (∞,1)-functor which exhibits $𝒟$ as an idempotent completion of? $𝒞$. Let $\kappa$ be a regular cardinal. Then the induced functor on (∞,1)-categories of ind-objects

${\mathrm{Ind}}_{\kappa }\left(f\right):{\mathrm{Ind}}_{\kappa }\left(𝒞\right)\to {\mathrm{Ind}}_{\kappa }\left(𝒟\right)$Ind_\kappa(f) \colon Ind_\kappa(\mathcal{C}) \to Ind_\kappa(\mathcal{D})

This is (Lurie, lemma 5.5.1.3).

###### Lemma

Let $L:𝒞\to 𝒟$ 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 $\left(\infty ,1\right)$-category and $p:K\to C$ a diagram in $C$, also the over (∞,1)-category ${C}_{/\mathrm{pp}}$ as well as the under-$\left(\infty ,1\right)$-category ${C}_{p/}$ are locally presentable.

###### Example

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

### Limits and colimits

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

###### Proposition

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

So consider first the case that $C=\mathrm{PSh}\left(D\right)$ is a presheaf category. Write

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

$F\prime :={\mathrm{Hom}}_{C}\left(-,f\right):\mathrm{PSh}\left(D{\right)}^{\mathrm{op}}\to \infty \mathrm{Grpd}$F' := Hom_{C}(-,f) : PSh(D)^{op} \to \infty Grpd

the functor represented by $f$.

We claim that $F\simeq F\prime$, which proves that $F$ is represented by $F\circ {j}^{\mathrm{op}}$: since both $F$ and $F\prime$ preserve limits (hence colimits as functors on $\mathrm{PSh}\left(D\right)$) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of $D$ that it is sufficient to show that $F\circ {j}^{\mathrm{op}}\simeq F\prime \circ {j}^{\mathrm{op}}$. 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 $\mathrm{PSh}\left(D\right)$. Let $L:\mathrm{PSh}\left(D\right)\to C$ be the left adjoint reflector. Since it respects all colimits, the composite

$F\circ {L}^{\mathrm{op}}:\mathrm{PSh}\left(D{\right)}^{\mathrm{op}}\stackrel{{L}^{\mathrm{op}}}{\to }{C}^{\mathrm{op}}\stackrel{F}{\to }\infty \mathrm{Grpd}$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 $X\in \mathrm{PSh}\left(D\right)$.

By the general properties of reflective sub-(∞,1)-categories, we have that $C$ is the full sub-(∞,1)-category of $\mathrm{PSh}\left(D\right)$ on those objects that are local objects with respect to the morphisms that $L$ sends to equivalences. But $X$, since it presents $F\circ {L}^{\mathrm{op}}$, is manifestly local in this sense and therefore also represents $F\circ {L}^{\mathrm{op}}{\mid }_{C}$. 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 $\left(\infty ,1\right)$-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 $j:C\to {\mathrm{PSh}}_{\left(\infty ,1\right)}\left(c\right)$. Since this is a full and faithful (∞,1)-functor it is sufficient to check that the limit computed in $\mathrm{PSh}\left(C\right)$ 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 $I\to \mathrm{PSh}\left(C\right)$ a diagram of limit-preserving functors that ${\mathrm{lim}}_{i}{F}_{i}$ is a functor that commutes with all limits, let $a:J\to C$ be a diagram and compute (verbatim as in ordinary category theory)

$\begin{array}{rl}\underset{j}{\mathrm{lim}}\left(\underset{i}{\mathrm{lim}}{F}_{i}\right)\left({a}_{j}\right)& \simeq \underset{j}{\mathrm{lim}}\left(\underset{i}{\mathrm{lim}}{F}_{i}\left({a}_{j}\right)\right)\\ & \simeq \underset{i}{\mathrm{lim}}\left(\underset{j}{\mathrm{lim}}{F}_{i}\left({a}_{j}\right)\right)\\ & \simeq \underset{i}{\mathrm{lim}}{F}_{i}\left(\mathrm{lim}{a}_{j}\right)\\ & \simeq \left(\underset{i}{\mathrm{lim}}{F}_{i}\right)\left(\mathrm{lim}{a}_{j}\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left(\infty ,1\right)$-categories presented by combinatorial simplicial model categories

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

###### Remark

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

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

$C\stackrel{←}{\to }\stackrel{\to }{←}\stackrel{←}{\to }\cdots D.$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 $\left(\infty ,1\right)$-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.

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

###### Proposition

For $C$ and $D$ locally presentable $\left(\infty ,1\right)$-categories, write ${\mathrm{Func}}^{L}\left(C,D\right)\subset \mathrm{Func}\left(C,D\right)$ for the full sub-$\left(\infty ,1\right)$-category on left-adjoint $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-category with finite products, the $\left(\infty ,1\right)$-category ${\mathrm{Alg}}_{\left(\infty ,1\right)}\left(C\right)$ of algebras over $C$ 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$↪$algebraic lattices$\simeq$ Porst’s theoremsubobject lattices in accessible reflective subcategories of presheaf categories
category theorytoposes$↪$locally presentable categories$\simeq$ Adámek-Rosický’s theoremaccessible reflective subcategories of presheaf categories$↪$accessible categories
model category theorymodel toposes$↪$combinatorial model categories$\simeq$ Dugger’s theoremleft Bousfield localization of global model structures on simplicial presheaves
(∞,1)-topos theory(∞,1)-toposes$↪$locally presentable (∞,1)-categories$\simeq$
Simpson’s theorem
accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories$↪$accessible (∞,1)-categories

## References

The theory of locally presentable $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-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 $\left(\infty ,1\right)$-categories as localizations of $\left(\infty ,1\right)$-presheaf categories is a variant is due to

Revised on October 16, 2012 11:14:33 by Urs Schreiber (131.174.190.249)