category theory

# Contents

## Idea

An ind-object of a category $\mathcal{C}$ is a formal filtered colimit of objects of $\mathcal{C}$. Here “formal” means that the colimit is taken in the category of presheaves of $\mathcal{C}$ (the free cocompletion of $\mathcal{C}$). The category of ind-objects of $\mathcal{C}$ is written $ind$-$\mathcal{C}$ or $Ind(\mathcal{C})$.

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 $\mathcal{C}$ are regarded to converge to an object in $Ind(\mathcal{C})$, even if that object does not exist in $\mathcal{C}$ itself. Standard examples where ind-objects are relevant are categories $\mathcal{C}$ whose objects are finite in some sense, such as finite sets or finite dimensional 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 $Ind(\mathcal{C})$ in terms of that of $\mathcal{C}$ 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 $ind$-$\mathcal{C}$ to be diagrams $F:D\to \mathcal{C}$ 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 $\mathcal{C}$). We identify an ordinary object of $\mathcal{C}$ with the corresponding diagram $1\to \mathcal{C}$. To see what the morphisms should be between $F:D\to \mathcal{C}$ and $G:E\to \mathcal{C}$, we stipulate that

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

Thus, we should have

\begin{aligned} ind\text{-}\mathcal{C}(F,G) &= ind\text{-}\mathcal{C}(colim_{d\in D} F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D}\; ind\text{-}\mathcal{C}(F d, colim_{e\in E} G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; ind\text{-}\mathcal{C}C(F d, G e)\\ &\cong lim_{d\in D} colim_{e\in E}\; \mathcal{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

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

### As filtered colimits of representable presheaves

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

there is a functor $\alpha : D \to \mathcal{C}$ (with $D$ possibly large) such that

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

(with $Y$ the Yoneda embedding).

###### Definition

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

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

for $D$ some filtered category.

Those for which $D$ may be chosen to be $\mathbb{N}^{\leq}$, i.e. those that arise as sequential colimits, are also called strict ind-objects.

###### Remark

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

For more equivalent characterizations see at accessible category – Definition.

###### Remark

Given that $[\mathcal{C}^{op},Set]$ is the free cocompletion of $\mathcal{C}$, $Ind(\mathcal{C})$ 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} [\mathcal{C}^{op},Set](X,Y) & \simeq [\mathcal{C}^{op}, Set](colim_{d \in D} Y F d, colim_{d' \in D'} Y G d) \\ & \simeq lim_{d \in D} [\mathcal{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 $PSh(\mathcal{C})$ are computed objectwise (see again properties at colimit) this is

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

## 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 $\mathcal{C}$ the category of finitely presented objects of some equationally defined structure, $ind\text{-}\mathcal{C}$ 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.
• A formal scheme is an ind-object in schemes.

## Properties

In the following we write $\underset{\longrightarrow}{\lim}^f$ for the “formal colimits” defining ind-objects. I.e. if $\alpha \colon \mathcal{I} \to \mathcal{C}$ is a small diagram and with $i \colon Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ the defining inclusion, then

$\underset{\longrightarrow}{\lim}^f (\alpha) \;\coloneqq\; \underset{\longrightarrow}{\lim} (i \circ \alpha ) \,.$

### The category of ind-objects

###### Proposition

If $\mathcal{C}$ is a locally small category then so is $Ind(\mathcal{C})$.

###### Proposition

$Ind(\mathcal{C})$ admits small filtered colimits and the defining inclusion $Ind(\mathcal{C}) \hookrightarrow PSh(\mathcal{C})$ commutes with these colimits.

The following says that morphisms between ind-objects are represented by natural transformation of the filtered diagrams that represent them, possibly up to restricting these diagrams first along a final functor.

###### Proposition

Let

1. $\mathcal{I}_1$ and $\mathcal{I}_2$ be two small filtered categories;

2. $\alpha_1 \colon \mathcal{I}_1 \longrightarrow \mathcal{C}$ and $\alpha_2 \colon \mathcal{I}_2 \longrightarrow \mathcal{C}$ be two functors;

3. $f \;\colon\; \underset{\longrightarrow}{\lim}^f \alpha_1 \longrightarrow \underset{\longrightarrow}{\lim}^f \alpha_2$

a morphism between their images in $Ind(\mathcal{C})$.

Then there exists

1. a small filtered category $K$

2. final functors $p_i \colon K \longrightarrow \mathcal{I}_i$

3. a natural transformation $\phi \;\colon\; \alpha_1 \circ p_1 \longrightarrow \alpha_2 \circ p_2$

such that the following diagram commutes

$\array{ \underset{\longrightarrow}{\lim}^f(\alpha_1 \circ p_1) &\overset{\underset{\longrightarrow}{\lim}^f \phi}{\longrightarrow}& \underset{\longrightarrow}{\lim}^f(\alpha_2 \circ p_2) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \underset{\longrightarrow}{\lim}^f \alpha_1 &\underset{f}{\longrightarrow}& \underset{\longrightarrow}{\lim}^f \alpha_2 } \,.$
###### Corollary

For each $f \colon A_1 \longrightarrow A_2$ a morphism in $Ind(\mathcal{C})$, then there exists

1. a small filtered category $\mathcal{I}$;

2. functors $\alpha_i \;\colon\; \mathcal{I} \to \mathcal{C}$ ($i \in \{1,2\}$);

3. a natural transformation $\phi \colon \alpha_1 \longrightarrow \alpha_2$

such that

$\array{ A_1 &\overset{f}{\longrightarrow}& A_2 \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ \underset{\longrightarrow}{\lim}^f \alpha_1 &\underset{\underset{\longrightarrow}{\lim}^f \phi }{\longrightarrow}& \underset{\longrightarrow}{\lim}^f \alpha_2 } \,.$
###### Proposition

The canonical inclusion $y \;\colon\; \mathcal{C} \hookrightarrow Ind(\mathcal{C})$ (factoring the Yoneda embedding) is right exact.

###### Proposition

Let $\mathcal{C}$ have all finite limits or all small limits. Then also $Ind(\mathcal{C})$ has all finite or small limits, respectively, and the canonical functor $y \;\colon\; \mathcal{C} \longrightarrow Ind(\mathcal{C})$ preserves these, respectively.

###### Proposition

If $\mathcal{C}$ has cokernels, then so does $Ind(\mathcal{C})$.

If $\mathcal{C}$ has finite colimit coproducts, then $Ind(\mathcal{C})$ has small coproducts.

If $\mathcal{C}$ has all finite colimits, then $Ind(\mathcal{C})$ has all small colimits.

### Recognition of Ind-objects

###### Proposition

A functor $F \colon \mathcal{C}^{op} \to Set$ is in $Ind(\mathcal{C})$ (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.

###### Proposition

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

### Functoriality

Ind-cocompletion is functorial – in fact an underlying 2-functor of a lax-idempotent 2-monad (KZ-monad). More in detail:

###### Proposition

Let $F \colon \mathcal{C}_1 \longrightarrow \mathcal{C}$ be a functor. Then there is a unique extension $Ind(F)$ of this functor to ind-objects, i.e. a commuting diagram

$\array{ \mathcal{C}_1 &\overset{F}{\longrightarrow}& \mathcal{C}_2 \\ \downarrow && \downarrow \\ Ind(\mathcal{C}_1) &\underset{Ind(F)}{\longrightarrow}& Ind(\mathcal{C}_2) } \,,$

such that

$Ind(F)( \underset{\longrightarrow}{\lim}^f \alpha ) \simeq \underset{\longrightarrow}{\lim}^f ( F \circ \alpha ) \,.$

Moreover,

1. this construction respects composition in that

$Ind(G \circ F ) \simeq Ind(G) \circ Ind(F)$
2. if $F$ is a faithful functor or fully faithful functor, then so is $Ind(F)$, respectively.

###### Proposition

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two small categories. By prop. 9 the two projections out of their product category induce a functor of the form

$Ind(\mathcal{C}_1 \times \mathcal{C}_2) \longrightarrow Ind(\mathcal{C}_1) \times Ind(\mathcal{C}_2) \,.$

This is an equivalence of categories.

### The case that $\mathcal{C}$ already admits filtered colimits

###### Proposition

Let $\mathcal{C}$ be a category which has all small filtered colimits. Then the canonical functor $\mathcal{C} \longrightarrow Ind(\mathcal{C})$ defines a reflective subcategory, i.e. it is a fully faithful functor with a left adjoint $L$

$\mathcal{C} \underoverset {\underset{}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\bot} Ind(\mathcal{C})$

which takes formal filtered colimits to actual filtered colimits in $\mathcal{C}$:

$L \;\colon\; \underset{\longrightarrow}{\lim}^f \alpha \mapsto \underset{\longrightarrow}{\lim} \alpha$
###### Proposition

Let $F \colon \mathcal{J} \longrightarrow \mathcal{C}$ be a functor such that

1. $F$ is a fully faithful functor;

2. $\mathcal{C}$ has all small filtered colimits;

3. for every object $J \in \mathcal{J}$ its image $F(J) \in \mathcal{C}$ is compact.

Then the composite

$Ind(\mathcal{J}) \overset{Ind(F)}{\longrightarrow} Ind(\mathcal{C}) \overset{L}{\longrightarrow} \mathcal{C}$

(with $Ind(F)$ from prop. 9 and $L$ from prop. 11) is a fully faithful functor.

###### Proposition

If $\mathcal{C}$ is a category such that

1. every object of $\mathcal{C}$ is a filtered colimit of compact objects;

2. $\mathcal{C}$ has all small filtered colimits

then the composite functor

$Ind(\mathcal{C}_{cpt}) \longrightarrow Ind(\mathcal{C}) \overset{L}{\longrightarrow} \mathcal{C}$

(from prop. 12, where $\mathcal{C}_{cpt} \hookrightarrow \mathcal{C}$ is the full subcategory of compact objects) is an equivalence of categories.

## 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 $F : \mathcal{C}^{op} \to Set$. This means that the forgetful functor $(Y,const_F) \to \mathcal{C}$ 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-objects are discussed in

• Michael Artin, Barry Mazur, appendix of Étale homotopy theory, Lecture Notes in Maths. 100, Springer-Verlag, Berlin 1969.

(in their dual guise as pro-objects)

The relation between the Ind-completion and the ideal completion in order theory is discussed in section 1 of