Model category theory
Producing new model structures
Presentation of -categories
for stable/spectrum objects
for stable -categories
for -sheaves / -stacks
This page means to give an introduction to the notion of locally presentable category, and its related notions in higher category theory and survey some fundamental properties.
Expected background of the reader:
Basic idea in category theory
The general idea is that a locally presentable category is a large category generated from small data: from small objects under small colimit.
Generation from generators
The notion of locally presentable category is, at least roughly, an analogue for categories of the notion of a finitely generated module.
An abelian group is called finitely generated if there is a finite subset
of the underlying set of , such that every element of is a sum of such generating elements.
Now, the categorification of “commutative sum” is colimit. Hence let now be a category with all small colimits.
We say a subclass of objects or equivalently the full subcategory on this subclass generates if every object in is a colimit of objects in , hence the colimit over a diagram of the form
As before, such a presentation is all the more useful the “smaller” the generating data is. In order to grasp the various aspects of the notion of “smallness” in category theory we need to recall the notion of regular cardinal.
The cardinality of a set is regular if every coproduct/disjoint union of sets of cardinality smaller than and indexed by a set of cardinality smaller than is itself of cardinality smaller than .
The smallest regular cardinal is ℵ: every finite union of finite sets is itself a finite set. (See the entry on regular cardinals for a discussion as to whether one might consider some finite cardinals as being `regular'.)
We can now speak of objects that are “-small sums” using the notion of -filtered colimits:
In an ℵ-filtered category every finite diagram has a cocone. This is equivalent to:
for every pair of objects there is a third objct such that both have a morphism to it;
for every pair of parallel morphisms there is a morphism out of their codomain such that the two composites are equal.
The tower diagram category
Using this we have the central definition now:
A crucial characterizing property of -filtered colimits is the following:
A colimit in Set is -filtered precisely if it commutes with all -small limits.
In particular a colimit in Set is filtered (meaning: ℵ-filtered) precisely if it commutes with all finite limits.
An object is a -compact object if it commutes with -filtered colimits, hence if for any -filtered diagram, the canonical function
is a bijection.
We say is a small object if it is -compact for some regular cardinal .
Locally presentable category: generated from colimits over small objects
Using this we can now say:
There are a bunch of equivalent reformulations of the notion of locally presentable category. One of the more important ones we again motivate first by analogy with presentable modules:
Generation exhibited by epimorphism from a free object
If an abelian group is generated by a set as in example 1, this means equivalently that there is an epimorphism
from the free abelian group generated by , hence the group obtained by forming formal sums of elements in . Here the epimorphism sends formal sums to actual sums in :
If a full subcategory generates under colimits as in defn. 1, then there is a functor
which sends formal colimits to actual colimits in
Here by construction preserves all colimits.
Therefore conversely, given a colimit-preserving functor we want to say that it locally presents if is “suitably epi”.
It turns out that “suitably epi” is to be the following:
With this notion we have the following analog of the familiar statement that an abelian group is generated by precisely if there is an epimorphism :
A category is locally presentable according to def. 6 precisely if it is an accessible localization, def. 7,
for some small category .
This is due to (Adámek-Rosický, prop 1.46).
Left exact localizations
Summary and overview
In summary the discussion above says that the notion of locally presentable categories sits in a sequence of notions as indicated in the row labeled “category theory” in the following table. The other rows are supposed to indicate that regarding a category as a (1,1)-category and simply varying in this story the parameters in “(n,r)-category” one obtains fairly straightforward analogs of the notion of locally presentable category in other fragments of higher category theory. These we discuss in more detail further below.
Locally presentable categories: Large categories whose objects arise from small generators under small relations.
|(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|
Basic idea in model category theory
Model structure on simplicial presheaves
The analog of a category of presheaves in model category theory is the model structure on simplicial presheaves, which we now briefly indicate.
Write sSet for the category of simplicial sets. Here we always regard this as equipped with the standard model structure on simplicial sets .
Left Bousfield localization
Given a model category and set of morphisms, the left Bousfield localization is the model structure with the same cofibrations and weak equivalences the -local morphisms.
Combinatorial model categories
The simple idea of the following definition is to say that the model category analog of locally presentable category is simply a model structure on a locally presentable category.
See at combinatorial model category - Dugger’s theorem.
Basic idea in -category theory
For and two (∞,1)-categories and two models as quasi-categories, an (∞,1)-functor is simply a homomorphism of simplicial set .
The (∞,1)-category of (∞,1)-functors as a quasi-category is simply the hom object of simplicial set
Localizations of -categories
The notions of adjoint functors, full and faithful functors etc. have straightforward, essentially verbatim generalizations to -categories:
A pair of (∞,1)-functors
is a pair of adjoint (∞,1)-functors, if there exists a unit transformation – a morphism in the (∞,1)-category of (∞,1)-functors – such that for all and the induced morphism
is an equivalence of ∞-groupoids.
An (∞,1)-functor is a full and faithful (∞,1)-functor if for all objects the component
is an equivalence of ∞-groupoids.
Locally presentable -categories
We have then the essentially verbatim analog of the situation for ordinary categories:
And the equivalent characterization is now as before
An (∞,1)-category is a locally presentable (∞,1)-category, def. 16, precisely if it is equivalent to localization, def. 15,
of an (∞,1)-category of (∞,1)-presheaves, def. 12, such that preserves -filtered (∞,1)-colimits for some regular cardinal .
This appears as Lurie, theorem 18.104.22.168, attributed there to Carlos Simpson.
As before, if a locally presentable -category arises as the localization of a left exact (∞,1)-functor, then it is an (∞,1)-topos.
Presentation by combinatorial model categories
There is a close match between the theory of combinatorial model categories and locally presentable (∞,1)-categories.
This is part of Lurie, theorem 22.214.171.124.
Accordingly, every simplicial Quillen adjunction between combinatorial model categories gives rise to a pair of adjoint (∞,1)-functors between the corresponding locally presentable -categories.
Hence a left Bousfield localization of a model structure on simplicial presheaves presents a corresponding localization of an (∞,1)-category of (∞,1)-presheaves to a locally presentable (∞,1)-category.
The standard textbook for locally presentable categories is
Decent accounts of combinatorial model categories include secton A.2.6 of
The standard text for locally presentable (∞,1)-categories is section 5 of Lurie.