equivalences in/of $(\infty,1)$-categories
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:
A locally presentable $(\infty,1)$-category is an accessible (∞,1)-category that admits all small (∞,1)-colimits.
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.
An (∞,1)-category $\mathcal{C}$ is called locally presentable if
it is accessible
it has all small (∞,1)-colimits.
That $\mathcal{C}$ is locally presentable is equivalent to each of the following equivalent characterizations.
$\mathcal{C}$ 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.
$\mathcal{C}$ 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
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 $K$).
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 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 fibrant and cofibrant objects.
$\mathcal{C}$ 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.
There exists a regular cardinal $\kappa$ such that $\mathcal{C}$ is $\kappa$-accessible and $C^\kappa$ admits $\kappa$-small colimits;
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, theorem 5.5.1.1, following (Simpson).
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 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.
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.
We indicate stepts in the proof of prop. 1.
Let $f \colon \mathcal{C} \to \mathcal{D}$ be an (∞,1)-functor which exhibits $\mathcal{D}$ as an idempotent completion $\mathcal{C}$. Let $\kappa$ be a regular cardinal. Then the induced functor on (∞,1)-categories of ind-objects
This is (Lurie, lemma 5.5.1.3).
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.
(…)
For $C$ a locally presentable $(\infty,1)$-category and $p : K \to C$ a diagram in $C$, also the over (∞,1)-category $C_{/pp}$ 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.
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.
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:
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.
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
for the precomposition of $F$ with the (∞,1)-Yoneda embedding. Then let
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
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:
A locally presentable $(\infty,1)$-category $C$ has all small limits.
This is HTT, prop. 5.5.2.4.
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)
By prop. 1 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 fibrant-cofibrant objects of $C$ (or, equivalently, the quasi-category associated to this simplicially enriched category).
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 $(\infty,1)$-categories precisely if there exists a chain of simplicial Quillen equivalence
This is Lurie, remark A.3.7.7.
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 $S$.
The basic example is:
∞Grpd is locally presentable.
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.
An (∞,1)-topos is precisely a locally presentable $(\infty,1)$-category where the localization functor also preserves finite limits.
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 .
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.
Locally presentable categories: Cocomplete possibly-large categories generated under filtered colimits by small generators under small relations. Equivalently, accessible localizations of free cocompletions. Accessible categories omit the cocompleteness requirement; toposes add the requirement of a left exact localization.
The theory of locally presentable $(\infty,1)$-categories was first implicitly conceived in terms of model category presentations in
The full intrinsic $(\infty,1)$-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 $(\infty,1)$-categories as localizations of $(\infty,1)$-presheaf categories is a variant is due to