nLab
locally presentable (infinity,1)-category

Idea

The notion of locally presentable $(\mathrm{\infty }\mathrm{,1})$-category is the analog in (∞,1)-category theory of locally presentable category in ordinary category theory:

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

a locally presentable $(\mathrm{\infty }\mathrm{,1})$-category $C$ is precisely

• a reflective (∞,1)-subcategory of an (∞,1)-category of (∞,1)-presheaves

  $$C\overleftarrow{\hookrightarrow }{\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(K);$$C \stackrel{\leftarrow}{\hookrightarrow} PSh_{(\infty,1)}(K) \,;

  such that the inclusion functor is an accessible (∞,1)-functor.

• equivalent to the full subcategory $A^{\circ }$ of a combinatorial simplicial model category $A$ on fibrant-cofibrant objects

  (where the Kan complex-enriched $A^{\circ }$ is regarded as an $(\mathrm{\infty }\mathrm{,1})$-category, for instance as a quasi-category after applying the homotopy coherent nerve).

If the left adjoint (∞,1)-functor to the full and faithful (∞,1)-functor $C\hookrightarrow {\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ also preserves finite limits, then the locally presentable $C$ is an (∞,1)-topos.

Warning on terminology. In HTT – where the notion has been introduced – the term presentable $(\mathrm{\infty }\mathrm{,1})$-category is used for what we call a locally presentable $(\mathrm{\infty }\mathrm{,1})$-category here, in order to be in line with the established terminology in ordinary category theory.

Presentation by simplicial model categories

Presentable $(\mathrm{\infty }\mathrm{,1})$-categories are precisely 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 fibrant-cofibrant objects of $C$

(Or, equivalently, the quasi-category associated to this simplicially enriched category).

Under this presentation, equivalence of presentable $(\mathrm{\infty }\mathrm{,1})$-categories corresponds precisely to spans of Quillen equivalences between presenting combinatorial simplicial model categories.

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

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

This is remark A.3.7.7 in HTT.

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 $(\mathrm{\infty }\mathrm{,1})$-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 $S$.

Presentation by conglomerates of objects in a small $(\mathrm{\infty }\mathrm{,1})$-category

From another perspective, the notion of a presentable (∞,1)-category is a means to handle large $(\mathrm{\infty }\mathrm{,1})$-categories in terms of small ones. It is is a slight refinement of the notion of an accessible (∞,1)-category.

A presentable $(\mathrm{\infty }\mathrm{,1})$-category is one which may be large, but can entirely be presented as an $(\mathrm{\infty }\mathrm{,1})$-category of “conglomerates of objects” in a small $(\mathrm{\infty }\mathrm{,1})$-category – precisely: that it is accessible but also admits all small colimits.

This means that it is desireable to get hold of presentable $(\mathrm{\infty }\mathrm{,1})$-categories. The following long list of equivalent definitions allows for many equivalent characterization of presentable $(\mathrm{\infty }\mathrm{,1})$-categories. In particular, all (∞,1)-categories of (∞,1)-sheaves are presentable.

Definition

Locally presentable $(\mathrm{\infty }\mathrm{,1})$-category

Definition

An (∞,1)-category $C$ is called locally presentable if it is accessible and admits small colimits.

Proposition

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

• $C$ is the localization of an (∞,1)-category of (∞,1)-presheaves:

  there exists a small $(\mathrm{\infty }\mathrm{,1})$-category $D$ and a functor $f:\mathrm{PSh}(D)\to C$ from the (∞,1)-category of (∞,1)-presheaves on $D$ with a fully faithful right adjoint.

  • (if in addition $f$ is left exact then $C$ is an (∞,1)-category of (∞,1)-sheaves on $C$)

• there exists a combinatorial simplicial model category $A$ and (with $C$ incarnated as a quasi-category) an equivalence $C\simeq N(A^{\circ })$ of $C$ with the homotopy coherent nerve of the full sSet-enriched subcategory of $A$ on fibrant and cofirant objects.

• $C$ is accessible and for every regular cardinal $\kappa$ the full subcategory $C^{\kappa }$ (…explanation to be filled in…) admits $\kappa$-small colimits;

• there exists a regular cardinal $\kappa$ such that $C$ is $\kappa$-accessible and $C^{\kappa }$ (explanation) admits $\kappa$-small colimits;

• there exists a regular cardinal $\kappa$, a small $(\mathrm{\infty }\mathrm{,1})$-category $D$ with $\kappa$-small colimits and an equivalence ${\mathrm{Ind}}_{\kappa }D\stackrel{\simeq }{\to }D$ of $D$ with the category of $\kappa$-ind-objects of $D$;

• $C$ is locally small, admits small colimits, and there exists a regular cardinal $\kappa$ and a small set $\mathrm{Cmptcs}$ of $\kappa$-compact objects of $C$ such that every object in $C$ is a colimit of a small diagram in the full subcategory on $\mathrm{Cmpcts}$.

The equivalent $(\mathrm{\infty }\mathrm{,1})$-categorical characterizations are originally due to Carlos Simpson. The whose theorem appears as HTT, theorem 5.5.1.1.

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

• Dan Dugger, Combinatorial model categories have presentations

The $(\mathrm{\infty }\mathrm{,1})$-category of presentable $(\mathrm{\infty }\mathrm{,1})$-categories

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

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

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

This is HTT, def. 5.5.3.1.

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

Properties

Limits and colimits

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

Proposition

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

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

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

$$F\prime :={\mathrm{Hom}}_C(-,f):\mathrm{PSh}(D{)}^{\mathrm{op}}\to \mathrm{\infty }\mathrm{Grpd}$$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 $\mathrm{PSh}(D)$) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of $D$ that it is sufficient to show that $F\circ j^{\mathrm{op}}\simeq F\prime \circ j^{\mathrm{op}}$. 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 $\mathrm{PSh}(D)$. Let $L:\mathrm{PSh}(D)\to C$ be the left adjoint reflector. Since it respects all colimits, the composite

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

By the general properties of reflective sub-(∞,1)-categories, we have that $C$ is the full sub-(∞,1)-category of $\mathrm{PSh}(D)$ on those objects that are local objects with respect to the morphisms that $L$ sends to equivalences. But $X$, since it presents $F\circ L^{\mathrm{op}}$, 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 $(\mathrm{\infty }\mathrm{,1})$-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 $j:C\to {\mathrm{PSh}}_{(\mathrm{\infty }\mathrm{,1})}(c)$. Since this is a full and faithful (∞,1)-functor it is sufficient to check that the limit computed in $\mathrm{PSh}(C)$ 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 $I\to \mathrm{PSh}(C)$ 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)

$$\begin{array}{rl}\underset{j}{\lim }(\underset{i}{\lim }{F}_i)(a_j)& \simeq \underset{j}{\lim }(\underset{i}{\lim }{F}_i(a_j))\\ & \simeq \underset{i}{\lim }(\underset{j}{\lim }{F}_i(a_j))\\ & \simeq \underset{i}{\lim }{F}_i(\lim a_j)\\ & \simeq (\underset{i}{\lim }{F}_i)(\lim a_j)\end{array}.$$\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} \,.

Examples

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

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

Proposition

For $C$ and $D$ locally presentable $(\mathrm{\infty }\mathrm{,1})$-categories, write ${\mathrm{Func}}^L(C,D)\subset \mathrm{Func}(C,D)$ for the full sub-$(\mathrm{\infty }\mathrm{,1})$-category on left-adjoint $(\mathrm{\infty }\mathrm{,1})$-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 $C$ a locally presentable $(\mathrm{\infty }\mathrm{,1})$-category and $p:K\to C$ a diagram in $C$, also the over (∞,1)-category $C_{/\mathrm{pp}}$ as well as the under-$(\mathrm{\infty }\mathrm{,1})$-category $C_{p/}$ are locally presentable.

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

Proposition

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

References

The theory of locally presentable $(\mathrm{\infty }\mathrm{,1})$-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 $\mathrm{\infty }$-pretopoi (arXiv:math/9903167)

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

• Jacob Lurie, Higher Topos Theory

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