category theory

# Contents

## Idea

An ind-object of a category $C$ is a formal filtered colimit of objects of $C$. The category of ind-objects of $C$ is written $ind$-$C$.

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

$c_1 \hookrightarrow c_2 \hookrightarrow c_3 \hookrightarrow c_4 \hookrightarrow \cdots$

of objects in $C$ are regarded to converge to an object in $ind C$, even if that object does not exist in $C$ itself. Standard examples where ind-objects are relevant are categories $C$ 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 $ind-C$ in terms of that of $C$ 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 $ind$-$C$ to be diagrams $F:D\to C$ where $D$ 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 $C$). We identify an ordinary object of $C$ with the corresponding diagram $1\to C$. To see what the morphisms should be between $F:D\to C$ and $G:E\to C$, we stipulate that

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

Thus, we should have

\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

$ind\text{-}C(F,G) \coloneqq lim_{d\in D} colim_{e\in E}\; C(F d, G e) \,.$

### As filtered colimits of representable presheaves

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

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

$X \simeq colim_{d \in D} Y(\alpha(d)) \,.$
###### Definition

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

$X \simeq colim_{d \in D} Y(\alpha(d))$

with $D$ a filtered category.

###### Remark

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

For more equivalent characterizations see at accessible category – Definition.

###### Definition

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

\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

$\cdots \simeq lim_{d \in D} (colim_{d' \in D'} Y G d')(F d) \,.$

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

$\cdots \simeq lim_{d \in D} colim_{d' \in D'} C(F d, G d') \,.$

## Examples

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

• For $C$ the category of finitely presented objects of some equationally defined structure, $ind\text{-}C$ 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

• If $C$ is a locally small category then so is $ind-C$.

• The inclusion $C \hookrightarrow ind\text{-}C$ is right exact.

• a functor $F : C^{op} \to Set$ is in $ind\text{-}C$ (i.e. is a filtered colimit of representables) precisely if the comma category $(Y,const_F)$ (with $Y$ the Yoneda embedding) is filtered and cofinally small.

• $ind\text{-}C$ admits small filtered colimits and the inclusion $ind\text{-}C\hookrightarrow PSh(C)$ commutes with these colimits.

• If $C$ admits finite colimits, then $ind\text{-}C$ is the full subcategory of the presheaf category $PSh(C)$ consisting of those functors $F : C^{op} \to Set$ such that $F$ is left exact and the comma category $(Y,F)$ (with $Y$ the Yoneda embedding) is cofinally small.

## In higher category theory

### In $(\infty,1)$-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 $(Y,const_F)$ is the category of elements of $F$, i.e. the pullback of the universal Set-bundle $U : Set_* \to Set$ along $F : C^{op} \to Set$. This means that the forgetful functor $(Y,const_F) \to C$ is the fibration classified by $F$.

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