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, as contrasted with “pro” in the dual notion for “projective system”.

Would it make sense to take this opportunity on nLab to give this a new modern name? Maybe call it “colimit-object” or “filtered colimit” or something? Something more consistent with the rest of the nLab might be nice. Just a thought. - Eric

Tim: As they are very special diagrams, I do not think that Eric’s suggestion is a good one. The pro and ind terminology was to replace inverse and direct system which was already being used in the 1930s (I think?) The ‘ind’ terminology is at least as old as 1960 (see Grothendeck’s seminar on descent.)

Mike: I think the terminology “ind-object” is too well-established to be changed. And I don’t really think there’s much reason to want to change it anyway; it may come historically from a terminology that we do not use, but it is unambiguous and the notion deserves its own name. In particular, it is not a (filtered) colimit; the whole point is that you don’t actually take the colimit (or alternately, you only take it inside the free colimit-completion).

Toby: Not to disparage ‘ind-object’, but I think that the term that Eric's looking for is ‘formal filtered colimit’.

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 finitely presentable 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 finitely presentable;

• 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 in 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.

Remarks

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.

Generalizations

in $(\mathrm{\infty }\mathrm{,1})$-catgeories

There is a notion of ind-object in an (infinity,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 (infinity,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

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Discussion

We had the following discussion on the comma category $(Y,{\mathrm{const}}_F)$.

David Roberts: I don’t think this is true, since the fibration classified by $F$ would be $(F,U)$ (or $(U,F)$, whichever is suitable) where $U:{\mathrm{Set}}_{*}\to \mathrm{Set}$ is the forgetful functor$=$universal set bundle. The $F$ in $(Y,F)$ is to be considered an object of ${\mathrm{Set}}^{C^{\mathrm{op}}}$, not as a functor $C^{\mathrm{op}}\to \mathrm{Set}$.

Urs: Ahm, let me see. So $(Y,F)$ (which, yes, should more precisely be written $(Y,{\mathrm{const}}_F)$) has as objects morphisms of presheaves $e:Y(c)\to F$, equivalently by Yoneda the objects are elements $e\in F(c)$.

Its morphisms from $e:Y(c)\to F$ to $e\prime :Y(c\prime )\to F$ are given by morphisms $f:c\to c\prime$ in $C$ such that $f^{*}e\prime =e$.

That seems to me to be the category which is also characterized as being the (strict) pullback of

$$\begin{array}{ccc}& & {\mathrm{Set}}_{*}\\ & & {\downarrow }^U\\ C^{\mathrm{op}}& \stackrel{F}{\to }& \mathrm{Set}\end{array}$$\array{ && Set_* \\ && \downarrow^{U} \\ C^{op} &\stackrel{F}{\to}& Set }

No?

Urs: Accordingly, I have now made the notation more precise by replacing ”$F$” by “const_F” in the above.