An ind-object of a category is a formal filtered colimit of objects of . Formal means that the colimit is taken in the category of presheaves. 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, and as contrasted with “pro” in the dual notion of pro-object corresponding to “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 one 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 several 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
the first step is by assumption that each object is a suitable colimit;
the second by the fact that the contravariant Hom sends colimits to limits (see properties of colimit);
the third by the assumption that each object is a compact object;
the last by the assumption that the embedding is a full and faithful functor.
So then one defines
there is a functor (with possibly large) such that
with a filtered category.
The functors belonging to under this definition — those which are filtered colimits of representables — have an equivalent characterization as the flat functors: those which “would preserve all finite colimits if had them”. In particular, if has finite colimits, then consists exactly of the finitely cocontinuous presheaves.
For more equivalent characterizations see at accessible category – Definition.
Given that is the free cocompletion of , - defined in this way is its “free cocompletion under filtered colimits.”
To compare with the first definition, notice that indeed the formula for the hom-sets is reproduced:
Generally we have
By the Yoneda lemma this is
Let FinVect be the category of finite-dimensional vector spaces (over some field). Let be an infinite-dimensional vector space. Then can be regarded as an object of 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.
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.
Ind-cocompletion is functorial – in fact an underlying 2-functor of a lax-idempotent 2-monad (KZ-monad)
There is a notion of ind-object in an (∞,1)-category.
With regard to the third of the properties listed above, notice that the comma category is the category of elements of , i.e. the pullback of the universal Set-bundle along . This means that the forgetful functor is the fibration classified by .
This is the starting point for the definition at ind-object in an (∞,1)-category.
ind-object / ind-object in an (∞,1)-category
Ind-categories were introduced in
and the dual notion of pro-object in
Ind-categories are discussed in
They are discussed in relation to generalisations in
See also the remarks at the beginning of section 5.3 of