An ind-object of a category is a formal filtered colimit of objects of . Here “formal” means that the colimit is taken in the category of presheaves of (the free cocompletion of ). The category of ind-objects of is written - or .
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 dimensional 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
- The embedding - should be full and faithful,
- each diagram should be the colimit of itself (considered as a diagram in - via the above embedding), and
- the objects of should be compact in -.
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
As filtered colimits of representable presheaves
Recall the co-Yoneda lemma that every presheaf is a colimit over representable presheaves:
there is a functor (with possibly large) such that
To compare with the first definition, notice that indeed the formula for the hom-sets is reproduced:
Generally we have
by the fact that the hom-functor sends colimits to limits in its first argument (see properties at colimit).
By the Yoneda lemma this is
Using that colimits in are computed objectwise (see again properties at colimit) 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.
A formal scheme is an ind-object in schemes.
In the following we write for the “formal colimits” defining ind-objects. I.e. if is a small diagram and with the defining inclusion, then
The category of ind-objects
admits small filtered colimits and the defining inclusion commutes with these colimits.
(e.g. KashiwaraSchapira 06, theorem 6.1.8)
The following says that morphisms between ind-objects are represented by natural transformation of the filtered diagrams that represent them, possibly up to restricting these diagrams first along a final functor.
and be two small filtered categories;
and be two functors;
a morphism between their images in .
Then there exists
a small filtered category
a natural transformation
such that the following diagram commutes
(e.g. KashiwaraSchapira 06, prop. 6.1.13, Artin-Mazur 69, appendix 3, prop. (3.1), corollary (3.2))
For each a morphism in , then there exists
a small filtered category ;
a natural transformation
(e.g. KashiwaraSchapira 06, corollary 6.1.14)
The canonical inclusion (factoring the Yoneda embedding) is right exact.
(e.g. KashiwaraSchapira 06, corollary 6.1.6)
Let have all finite limits or all small limits. Then also has all finite or small limits, respectively, and the canonical functor preserves these, respectively.
(e.g. KashiwaraSchapira 06, corollary 6.1.17)
(e.g. KashiwaraSchapira 06, prop. 6.1.18)
Recognition of Ind-objects
Ind-cocompletion is functorial – in fact an underlying 2-functor of a lax-idempotent 2-monad (KZ-monad). More in detail:
Let be a functor. Then there is a unique extension of this functor to ind-objects, i.e. a commuting diagram
this construction respects composition in that
if is a faithful functor or fully faithful functor, then so is , respectively.
(e.g. KashiwaraSchapira 06, prop.6.1.9-6.1.11)
(KashiwaraSchapira 06, prop. 6.1.12)
The case that already admits filtered colimits
(KashiwaraSchapira 06, chapter 6.3)
Let be a category which has all small filtered colimits. Then the canonical functor defines a reflective subcategory, i.e. it is a fully faithful functor with a left adjoint
which takes formal filtered colimits to actual filtered colimits in :
(KashiwaraSchapira 06, prop. 6.3.1)
Let be a functor such that
is a fully faithful functor;
has all small filtered colimits;
for every object its image is compact.
Then the composite
(with from prop. 9 and from prop. 11) is a fully faithful functor.
(KashiwaraSchapira 06, prop. 6.3.4)
(KashiwaraSchapira 06, corollary 6.3.5)
In higher category theory
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-categories were introduced in
and the dual notion of pro-object in
- A. Grothendieck, Techniques de déscente et théorèmes d’existence en géométrie algébrique, II: le théorème d’existence en théorie formelle des modules, Seminaire Bourbaki 195, 1960, (pdf).
Ind-objects are discussed in
- Michael Artin, Barry Mazur, appendix of Étale homotopy theory, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.
(in their dual guise as pro-objects)
The relation between the Ind-completion and the ideal completion in order theory is discussed in section 1 of
They are discussed in relation to generalisations in
See also the remarks at the beginning of section 5.3 of