nLab
ind-object

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 for “projective system”.

Recalling the nature of filtered colimits, this means that in particular chains of inclusions

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

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

$$\mathrm{ind}\text{-}C(F,G):=\cong {\lim }_{d\in D}{\mathrm{colim}}_{e\in E}C(Fd,Ge).$$ind\text{-}C(F,G) := \cong 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 $X\in \mathrm{PSh}(C)$ is a colimit over representable presheaves:

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

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

Definition

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

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

with $D$ a filtered category.

Remark Given that $[C^{\mathrm{op}},\mathrm{Set}]$ 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 s reproduced:

Generally we have

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

$$\cdots \simeq {\lim }_{d\in D}({\mathrm{colim}}_{d\prime \in D\prime }YGd\prime )(Fd).$$\cdots \simeq lim_{d \in D} (colim_{d' \in D'} Y G d')(F d) \,.

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

$$\cdots \simeq {\lim }_{d\in D}{\mathrm{colim}}_{d\prime \in D\prime }C(Fd,Gd\prime ).$$\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 ${\mathrm{colim}}_{V\prime \hookrightarrow V}Y(V\prime )$ over the filtered category whose objects are inclusions $V\prime \hookrightarrow V$ of finite dimensional vector spaces $V\prime$ into $V$ of the representables $Y(V\prime ):{\mathrm{FinVect}}^{\mathrm{op}}\to \mathrm{Set}$ ($Y$ 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 $C\hookrightarrow \mathrm{ind}\text{-}C$ 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 $(Y,{\mathrm{const}}_F)$ (with $Y$ the Yoneda embedding) is filtered and cofinally small.

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

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

Applications

• One important use of categories of ind-objects is in abelian sheaf-theory: for every small abelian category $C$ the category $\mathrm{ind}\text{-}C$ is a Grothendieck category and hence a good coefficient object for abelian sheaf cohomology.

In higher category theory

In $(\mathrm{\infty },1)$-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 $(Y,{\mathrm{const}}_F)$ 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 $(Y,{\mathrm{const}}_F)\to C$ is the fibration classified by $F$.

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

Related concepts

• ind-object / ind-object in an (∞,1)-category

• pro-object / pro-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.

See also the remarks at the beginning of section 5.3 of

• Jacob Lurie, Higher Topos Theory