nLab
ind-object

Contents

Idea

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, 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

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 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 indCind-C in terms of that of CC below). Such large categories equivalent to ind-categories are therefore called accessible categories.

Definition

There are several 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 compact 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}

Here

  • 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

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

As filtered colimits of representable presheaves

Recall the co-Yoneda lemma that every presheaf XPSh(C)X \in PSh(C) is a colimit over representable presheaves:

there is some functor α:DC\alpha : D \to C such that

Xcolim dDY(α(d)). X \simeq colim_{d \in D} Y(\alpha(d)) \,.
Definition

Let ind-CPSh(C)ind\text{-}C \subset PSh(C) be the full subcategory of the presheaf category PSh(C)=[C op,Set]PSh(C) = [C^{op},Set] on those functors which are filtered colimits of representables, i.e. those for which

Xcolim dDY(α(d)) X \simeq colim_{d \in D} Y(\alpha(d))

with DD a filtered category.

Remark

The functors C opSetC^{op}\to Set belonging to ind-Cind\text{-}C 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 CC had them”. In particular, if CC has finite colimits, then ind-Cind\text{-}C consists exactly of the finitely cocontinuous presheaves.

For more equivalent characterizations see at accessible category – Definition.

Definition

Given that [C op,Set][C^{op},Set] is the free cocompletion of CC, indind-CC 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

[C op,Set](X,Y) [C op,Set](colim dDYFd,colim dDYGd) lim dD[C op,Set](YFd,colim dDYGd) \begin{aligned} [C^{op},Set](X,Y) & \simeq [C^{op}, Set](colim_{d \in D} Y F d, colim_{d' \in D'} Y G d) \\ & \simeq lim_{d \in D} [C^{op}, Set]( Y F d, colim_{d' \in D'} Y G d) \end{aligned}

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

lim dD(colim dDYGd)(Fd). \cdots \simeq lim_{d \in D} (colim_{d' \in D'} Y G d')(F d) \,.

Using that colimits in PSh(C)PSh(C) are computed objectwise (see again properties at colimit) this is

lim dDcolim dDC(Fd,Gd). \cdots \simeq lim_{d \in D} colim_{d' \in D'} C(F d, G d') \,.

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 category Grp of groups is the ind-category of the category of finitely generated groups.

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

Properties

Applications

In higher category theory

In (,1)(\infty,1)-categories

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 (Y,const F)(Y,const_F) is the category of elements of FF, i.e. the pullback of the universal Set-bundle U:Set *SetU : Set_* \to Set along F:C opSetF : C^{op} \to Set. This means that the forgetful functor (Y,const F)C (Y,const_F) \to C is the fibration classified by FF.

This is the starting point for the definition at ind-object in an (∞,1)-category.

References

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-categories are discussed in

They are discussed in relation to generalisations in

See also the remarks at the beginning of section 5.3 of

Revised on February 16, 2014 08:26:38 by Tim Porter (2.26.28.101)