Schreiber
accessing big categories -- filtered colimits and ind-objects

previous: universal constructions – adjunction, limit and Kan extension

home: sheaves and stacks

next: monoidal and enriched categories


  • we had seen that the category of presheaves on a small category CC is the free cocompletion – the free closure under colimits of CC.

  • in particular, every presheaf is a small colimit of representables, i.e. of objects of CC.

  • 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 CC and PSh(C)PSh(C):

    • this is the category ind-Cind\text{-}C of ind-objects of CC: these are presheaves which are not arbitrary but just filtered colimits of representables.
  • 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.

Motivation – One more example: colimits over posets

  • if DD is a poset, then the colimit over DD is the supremum over the F(d)F(d) with respect to (F(d)F(d))(F(d)F()F(d))(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\to} F(d'));

The generalization of this is where the term “imit” for categorical (co)limit (probably) originates from: where DD a filtered category.

Filtered categories

A filtered category is a category DD in which any finite diagram has a cocone. That is, for any finite category KK and any functor G:KDG:K \to D, there exists an object dDd\in D and a natural transformation Gconst dG\to const_d.

This can be rephrased in more elementary terms by saying that:

  • There exists an object of DD (the case when K=K=\emptyset)
  • For any two objects d 1,d 2Dd_1,d_2\in D, there exists an object d 3Dd_3\in D and morphisms d 1d 3d_1\to d_3 and d 2d 3d_2\to d_3.
  • For any two parallel morphisms f,g:d 1d 2f,g:d_1\to d_2 in DD, there exists a morphism h:d 2d 3h:d_2\to d_3 such that hf=hgh f = h g.

One may think of (dfd)(F(d)F(f)F(d)(d \stackrel{f}{\to} d') \mapsto (F(d) \stackrel{F(f)}{\to} F(d') as witnessing that F(d)F(d) is “smaller than” or “sitting inside” F(d)F(d') in some sense, and colimFcolim F is then the “largest” of all these objects, the limiting object.

Ind-Objects

An ind-object of a category CC is a “formal filtered colimit” of objects of CC. The category of ind-objects of CC is written indind-CC.

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

c 1c 2c 3c 4 c_1 \hookrightarrow c_2 \hookrightarrow c_3 \hookrightarrow c_4 \hookrightarrow \cdots

of objects in CC are regarded to converge to an object in indCind C, even if that object does not exist in CC itself. Standard examples where ind-objects are relevant are categories CC 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 indCind-C in terms of that of CC below). Such large categories equivalent to ind-categories are therefore called accessible categories.

Definition

There are two equivalent ways to define ind-objects.

as diagrams

One definition is to define the objects of indind-CC to be diagrams F:DCF:D\to C where DD 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 CC) We identify an ordinary object of CC with the corresponding diagram 1C1\to C. To see what the morphisms should be between F:DCF:D\to C and G:ECG:E\to C, we stipulate that

  1. The embedding CindC\to ind-CC should be full and faithful,
  2. each diagram F:DCF:D\to C should be the colimit of itself (considered as a diagram in indind-CC via the above embedding), and
  3. the objects of CC should be finitely presentable in indind-CC.

Thus, we should have

ind-C(F,G) =ind-C(colim dDFd,colim eEGe) lim dDind-C(Fd,colim eEGe) lim dDcolim eEind-C(Fd,Ge) lim dDcolim eEC(Fd,Ge) \begin{aligned} ind\text{-}C(F,G) &= ind\text{-}C(colim_{d\in D} F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D}\; ind\text{-}C(F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; ind\text{-}C(F d, G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; C(F d, G e) \end{aligned}

So then one defines

ind-C(F,G):=lim dDcolim eEC(Fd,Ge). ind\text{-}C(F,G) := \cong lim_{d\in D} colim_{e\in E}\; C(F d, G e) \,.

as filtered colimits in presheaves

Another, equivalent, definition is to let indind-CC be the full subcategory of the presheaf category [C op,Set][C^{op},Set] determined by those functors which are filtered colimits of representables. This is reasonable since [C op,Set][C^{op},Set] is the free cocompletion of CC, so indind-CC defined in this way is its “free cocompletion under filtered colimits.”

Examples

  • Let FinVectFinVect be the category of finite-dimensional vector spaces (over some field). Let VV be an infinite-dimensional vector space. Then VV can be regarded as an object of indFinVectind-FinVect as the colimit colim VVY(V)colim_{V' \hookrightarrow V} Y(V') over the filtered category whose objects are inclusions VVV' \hookrightarrow V of finite dimensional vector spaces VV' into VV of the representables Y(V):FinVect opSetY(V') : FinVect^{op} \to Set (YY is the Yoneda embedding).

  • For CC the category of finitely presented objects of some equationally defined structure, ind-Cind\text{-}C is the category of all these structures.

    • The catgeory Grp of groups is the ind-category of the category of finitely generated groups.

      • The catgeory Ab of abelian groups is the ind-category of the category of finitely generated abelian groups.

Properties

Last revised on May 29, 2012 at 22:04:00. See the history of this page for a list of all contributions to it.