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 $\mathrm{ind}$-$C$.

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

${c}_{1}↪{c}_{2}↪{c}_{3}↪{c}_{4}↪\cdots$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 $\mathrm{ind}C$, 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 $\mathrm{ind}-C$ 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 $\mathrm{ind}$-$C$ 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 $C\to \mathrm{ind}$-$C$ should be full and faithful,
2. each diagram $F:D\to C$ should be the colimit of itself (considered as a diagram in $\mathrm{ind}$-$C$ via the above embedding), and
3. the objects of $C$ should be compact in $\mathrm{ind}$-$C$.

Thus, we should have

$\begin{array}{rl}\mathrm{ind}\text{-}C\left(F,G\right)& =\mathrm{ind}\text{-}C\left({\mathrm{colim}}_{d\in D}Fd,{\mathrm{colim}}_{e\in E}Ge\right)\\ & \cong {\mathrm{lim}}_{d\in D}\phantom{\rule{thickmathspace}{0ex}}\mathrm{ind}\text{-}C\left(Fd,{\mathrm{colim}}_{e\in E}Ge\right)\\ & \cong {\mathrm{lim}}_{d\in D}{\mathrm{colim}}_{e\in E}\phantom{\rule{thickmathspace}{0ex}}\mathrm{ind}\text{-}C\left(Fd,Ge\right)\\ & \cong {\mathrm{lim}}_{d\in D}{\mathrm{colim}}_{e\in E}\phantom{\rule{thickmathspace}{0ex}}C\left(Fd,Ge\right)\end{array}$\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

$\mathrm{ind}\text{-}C\left(F,G\right)≔{\mathrm{lim}}_{d\in D}{\mathrm{colim}}_{e\in E}\phantom{\rule{thickmathspace}{0ex}}C\left(Fd,Ge\right)\phantom{\rule{thinmathspace}{0ex}}.$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 $X\in \mathrm{PSh}\left(C\right)$ is a colimit over representable presheaves:

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

$X\simeq {\mathrm{colim}}_{d\in D}Y\left(\alpha \left(d\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$X \simeq colim_{d \in D} Y(\alpha(d)) \,.
Definition

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

$X\simeq {\mathrm{colim}}_{d\in D}Y\left(\alpha \left(d\right)\right)$X \simeq colim_{d \in D} Y(\alpha(d))

with $D$ a filtered category.

Remark Given that $\left[{C}^{\mathrm{op}},\mathrm{Set}\right]$ is the free cocompletion of $C$, $\mathrm{ind}$-$C$ 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

$\begin{array}{rl}\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\left(X,Y\right)& \simeq \left[{C}^{\mathrm{op}},\mathrm{Set}\right]\left({\mathrm{colim}}_{d\in D}YFd,{\mathrm{colim}}_{d\prime \in D\prime }YGd\right)\\ & \simeq {\mathrm{lim}}_{d\in D}\left[{C}^{\mathrm{op}},\mathrm{Set}\right]\left(YFd,{\mathrm{colim}}_{d\prime \in D\prime }YGd\right)\end{array}$\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

$\cdots \simeq {\mathrm{lim}}_{d\in D}\left({\mathrm{colim}}_{d\prime \in D\prime }YGd\prime \right)\left(Fd\right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq lim_{d \in D} (colim_{d' \in D'} Y G d')(F d) \,.

Using that colimits in $\mathrm{PSh}\left(C\right)$ are computed objectwise (see again properties at colimit) this is

$\cdots \simeq {\mathrm{lim}}_{d\in D}{\mathrm{colim}}_{d\prime \in D\prime }C\left(Fd,Gd\prime \right)\phantom{\rule{thinmathspace}{0ex}}.$\cdots \simeq lim_{d \in D} colim_{d' \in D'} C(F d, G d') \,.

Examples

• Let $\mathrm{FinVect}$ 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 $\mathrm{ind}-\mathrm{FinVect}$ as the colimit ${\mathrm{colim}}_{V\prime ↪V}Y\left(V\prime \right)$ over the filtered category whose objects are inclusions $V\prime ↪V$ of finite dimensional vector spaces $V\prime$ into $V$ of the representables $Y\left(V\prime \right):{\mathrm{FinVect}}^{\mathrm{op}}\to \mathrm{Set}$ ($Y$ is the Yoneda embedding).

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

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

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

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

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

In higher category theory

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

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

References

Ind-categories are discussed in

• Kashiwara-Schapira, Categories and Sheaves, section 6

• Grothendieck et al. SGA4.I.6 djvu file

• 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.