objects such that commutes with certain colimits
This says equivalently that a presentable category is a reflective localization of a category of presheaves over . Since here is the free colimit completion of and the localization imposes relations, this is a presentation of by generators and relations, hence the name (locally) presentable category.
See also at locally presentable categories - introduction.
There are many equivalent characterizations of locally presentable categories. The following is one of the most intuitive, equivalent characterizations are discussed below.
(locally presentable category)
A category is called locally presentable if
is a locally small category;
has all small colimits;
(meaning that every object of may be written as a colimit over a diagram with objects in );
every object in is a small object (assuming 3, this is equivalent to the assertion that every object in is small).
The locally in locally presentable category refers to the fact that it is the objects that are presentable, not the category as such.
For instance, consider the notion of “locally finitely presentable category”, def. 3 below, in which the generating set consists of finitely presentable objects, i.e. -small ones. If one dropped the word “locally” then one would get the notion “finitely presentable category” which means something completely different, namely a finitely presentable (-small) object of Cat.
Since a small object is one which is -compact for some , and any -compact object is also -compact for any , it follows that there exists some such that every object of the colimit-generating set is -compact.
This provides a “stratification” of the class of locally presentable categories, as follows.
(locally -presentable category)
Thus, a locally presentable category is one which is locally -presentable for some regular cardinal (hence also for every ). In fact, in this case the fourth condition is redundant; once we know that there is a colimit-generating set consisting of -compact objects, it follows automatically that every object is -compact for some (though there is no uniform upper bound on the required size of ). Moreover, colimit-generation is also stronger than necessary; it suffices to have a strong generator consisting of small objects.
There are various equivalent characterizations of locally presentable categories.
(as limit sketches)
Locally presentable categories are precisely the categories of models of limit-sketches.
This is (Adamek-Rosicky, corollary 1.52).
Restricted to locally finitely presentable categories this becomes:
For the more detailed statement see below at Gabriel-Ulmer duality. Equivalently this says that:
(as accessible reflective subcategories of presheaves)
of categories of presheaves on some category .
This appears as (Adámek-Rosický, prop 1.46).
There is an equivalence of 2-categories
A slice category of a locally presentable category is again locally presentable.
This appears for instance as (Centazzo-Rosický-Vitale, remark 3).
We list examples of locally finitely presentable categories, def. 3.
Hence a a set of generators that exhibits as a locally finitely complete category is given by the set containing one finite set of cardinality for all .
More generally still, if is locally finitely presentable and is small, then is locally finitely presentable.
The category of algebras of a Lawvere theory, for example Grp, is locally finitely presentable. A -algebra is finitely presented if and only if the hom-functor preserves filtered colimits, and any -algebra can be expressed as a filtered colimit of finitely presented algebras.
The category FinSet of finite sets is not locally finitely presentable, as it does not have all countable colimits.
The category of fields and field homomorphisms is not locally presentable, as it does not have all binary coproducts (for instance, there are none between fields of differing characteristics).
Top is not locally finitely presentable.
Every sheaf topos is locally presentable.
The main ingredient of a direct proof is:
If is an accessible monad (a monad whose underlying functor is an accessible functor) on a locally presentable category , then the category of algebras over the monad is locally presentable. In particular, if is locally presentable and is a reflective subcategory, then is locally presentable if is accessible.
This appears in (Adamek-Rosicky, 2.78).
This is actually somewhat subtle and gets into some transfinite combinatorics, from what I can gather.
Given a class of morphisms in a locally presentable category, the answer to the orthogonal subcategory problem for is affirmative if is small, and is affirmative for any class assuming the large cardinal axiom known as Vopenka's principle.
Another notion of “presentable category” is that of an equationally presentable category.
Locally presentable categories are a special case of locally bounded categories.
|(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 definition is due to
The standard textbook is
More details are in
Some further discussion is in
See also section A.1.1 of
where locally presentable categories are called just presentable categories.