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we had seen that the category of presheaves on a small category is the free cocompletion – the free closure under colimits of .
in particular, every presheaf is a small colimit of representables, i.e. of objects of .
for many applications, in particular for obtaining Grothendieck categories as coefficients of abelian sheaves, it is useful or necessary to have a category in between and :
filtered colimits generalize the notion of “limiting objects”, in the original sense: these are like “infinit objects” which are asymptotic to chains of inclusions of “finite object”s.
The generalization of this is where the term “imit” for categorical (co)limit (probably) originates from: where a filtered category.
A filtered category is a category in which any finite diagram has a cocone. That is, for any finite category and any functor , there exists an object and a natural transformation .
This can be rephrased in more elementary terms by saying that:
One may think of as witnessing that is “smaller than” or “sitting inside” in some sense, and is then the “largest” of all these objects, the limiting object.
An ind-object of a category is a “formal filtered colimit” of objects of . The category of ind-objects of is written -.
Here, “ind” is short for “inductive system”, as in the inductive systems used to define directed colimits, as contrasted with “pro” in the dual notion for “projective system”.
Recalling the nature of filtered colimits, this means that in particular chains of inclusions
of objects in are regarded to converge to an object in , even if that object does not exist in itself. Standard examples where ind-objects are relevant are categories whose objects are finite in some sense, such as finite sets or finite vector spaces. Their ind-categories contain then also the infinite versions of these objects as limits of sequences of inclusions of finite objects of ever increasing size.
Moreover, ind-categories allow to handle “big things in terms of small things” also in another important sense: many large categories are actually (equivalent to) ind-categories of small categories. This means that, while large, they are for all practical purposes controlled by a small category (see the description of the hom-set of in terms of that of below). Such large categories equivalent to ind-categories are therefore called accessible categories.
There are two equivalent ways to define ind-objects.
One definition is to define the objects of - to be diagrams where is a small filtered category.
The idea is to think of these diagrams as being the placeholder for the colimit over them (possibly non-existent in ) We identify an ordinary object of with the corresponding diagram . To see what the morphisms should be between and , we stipulate that
Thus, we should have
So then one defines
Another, equivalent, definition is to let - be the full subcategory of the presheaf category determined by those functors which are filtered colimits of representables. This is reasonable since is the free cocompletion of , so - defined in this way is its “free cocompletion under filtered colimits.”
Let FinDimVect be the category of finite-dimensional vector spaces (over some field). Let be an infinite-dimensional vector space. Then can be regarded as an ind-object as the colimit over the filtered category whose objects are inclusions of finite dimensional vector spaces into of the representables ( is the Yoneda embedding).
For the category of finitely presented objects of some equationally defined structure, is the category of all these structures.
If is a locally small category then so is .
The inclusion is right exact.
a functor is in (i.e. is a filtered colimit of representables) precisely if the comma category (with the Yoneda embedding) is filtered and cofinally small.
admits small filtered colimits and the inclusion commutes with these colimits.
If admits finite colimits, then is the full subcategory of the presheaf category consisting of those functors such that is left exact and the comma category (with the Yoneda embedding) is cofinally small.
Last revised on March 30, 2023 at 15:09:05. See the history of this page for a list of all contributions to it.