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:
The locally presentable -categories are precisely the accessibly embedded localizations/reflections of an (∞,1)-category of (∞,1)-presheaves. In particular, if the reflector of this reflection is a left exact (∞,1)-functor, then is an (∞,1)-topos.
See also at locally presentable categories - introduction.
Warning on terminology. In Lurie the term presentable -category is used for what we call a locally presentable -category here, in order to be in line with the established terminology of locally presentable category in ordinary category theory.
An (∞,1)-category is called locally presentable if
That is locally presentable is equivalent to each of the following equivalent characterizations.
More explicitly: with incarnated as a quasi-category there is equivalence of quasi-categories of with the homotopy coherent nerve of the full sSet-enriched subcategory of on fibrant and cofibrant objects.
That localizations 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 -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.
those objects that are locally presentable -categories;
those morphisms that are colimit-preserving (∞,1)-functors.
This is Lurie, def. 184.108.40.206.
This -category 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.
This is (Lurie, lemma 220.127.116.11).
Let be an (∞,1)-functor between (∞,1)-categories which have -filtered (∞,1)-colimits, and let be a right adjoint (∞,1)-functor of . If preserves -filtered (∞,1)-colimits then preserves -compact objects.
This is Lurie, lemma 18.104.22.168.
This is HTT, prop. 22.214.171.124, prop. 126.96.36.199.
Since Pr(∞,1)Cat admits all small limits, we obtain new locally presentable -categories by forming limits over given ones. In particular the product of locally presentable -categories is again locally presentable.
In the first definition of locally presentable -category above only the existence of colimits is postulated. An important fact is that it follows automatically that also all small limits exist:
This is HTT, prop. 188.8.131.52.
We need to prove that a limit-preserving functor is representable. By the above characterizations we know that is an accessible localization of a presheaf category.
So consider first the case that is a presheaf category. Write
for the precomposition of with the (∞,1)-Yoneda embedding. Then let
the functor represented by .
We claim that , which proves that is represented by : since both and preserve limits (hence colimits as functors on ) it follows from the fact that the Yoneda embedding exhibits the universal co-completion of that it is sufficient to show that . But this is the case precisely by the statement of the full (∞,1)-Yoneda lemma.
respects all limits. By the above it is therefore represented by some object .
By the general properties of reflective sub-(∞,1)-categories, we have that is the full sub-(∞,1)-category of on those objects that are local objects with respect to the morphisms that sends to equivalences. But , since it presents , is manifestly local in this sense and therefore also represents . But on the functor is equivalent to the identity, so that this is equivlent to .
This statement has the following important consequence:
A locally presentable -category has all small limits.
This is HTT, prop. 184.108.40.206.
We may compute the limit after applying the (∞,1)-Yoneda embedding . Since this is a full and faithful (∞,1)-functor it is sufficient to check that the limit computed in lands in the essential image of . 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 a diagram of limit-preserving functors that is a functor that commutes with all limits, let be a diagram and compute (verbatim as in ordinary category theory)
By prop. 1 locally presentable -categories are equivalently those (∞,1)-categories which are presented by a combinatorial simplicial model category in that they are the full simplicial subcategory on fibrant-cofibrant objects of (or, equivalently, the quasi-category associated to this simplicially enriched category).
Under this presentation, equivalence of (∞,1)-categories between locally presentable -categories corresponds to zigzags of Quillen equivalences between presenting combinatorial simplicial model categories:
and are equivalent as -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 -categories are indeed handled in terms of such a presentation by a simplicial model category.
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 -groupoid. This is clearly compact, and hence generates ∞Grpd.
For and locally presentable -categories, write for the full sub--category on left-adjoint -functors. This is itself locally presentable
This is HTT, prop 220.127.116.11
Notice that this makes the symmetric monoidal (∞,1)-category of presentable (∞,1)-categories closed .
|(n,r)-categories…||satisfying Giraud's axioms||inclusion of left exact localizations||generated under colimits from small objects||localization of free cocompletion||generated under filtered colimits from small objects|
|(0,1)-category theory||(0,1)-toposes||algebraic lattices||Porst’s theorem||subobject lattices in accessible reflective subcategories of presheaf categories|
|category theory||toposes||locally presentable categories||Adámek-Rosický’s theorem||accessible reflective subcategories of presheaf categories||accessible categories|
|model category theory||model toposes||combinatorial model categories||Dugger’s theorem||left Bousfield localization of global model structures on simplicial presheaves|
|(∞,1)-topos theory||(∞,1)-toposes||locally presentable (∞,1)-categories|| |
|accessible reflective sub-(∞,1)-categories of (∞,1)-presheaf (∞,1)-categories||accessible (∞,1)-categories|
The theory of locally presentable -categories was first implicitly conceived in terms of model category presentations in
The full intrinsic -categorical theory appears in section 5
The statement of Dugger’s theorem of which the characterization of locally presentable -categories as localizations of -presheaf categories is a variant is due to