# Schreiber accessing big categories -- filtered colimits and ind-objects

home: sheaves and stacks

• we had seen that the category of presheaves on a small category $C$ is the free cocompletion – the free closure under colimits of $C$.

• in particular, every presheaf is a small colimit of representables, i.e. of objects of $C$.

• for many applications, in particular for obtaining Grothendieck categories as coefficients of abelian sheaves, it is useful or necessary to have a category in between $C$ and $PSh(C)$:

• this is the category $ind\text{-}C$ of ind-objects of $C$: these are presheaves which are not arbitrary but just filtered colimits of representables.
• filtered colimits generalize the notion of “limiting objects”, in the original sense: these are like “infinit objects” which are asymptotic to chains of inclusions of “finite object”s.

# Motivation – One more example: colimits over posets

• if $D$ is a poset, then the colimit over $D$ is the supremum over the $F(d)$ with respect to $(F(d) \leq F(d')) \Leftrightarrow (F(d) \stackrel{F(\leq)}{\to} F(d'))$;

The generalization of this is where the term “imit” for categorical (co)limit (probably) originates from: where $D$ a filtered category.

# Filtered categories

A filtered category is a category $D$ in which any finite diagram has a cocone. That is, for any finite category $K$ and any functor $G:K \to D$, there exists an object $d\in D$ and a natural transformation $G\to const_d$.

This can be rephrased in more elementary terms by saying that:

• There exists an object of $D$ (the case when $K=\emptyset$)
• For any two objects $d_1,d_2\in D$, there exists an object $d_3\in D$ and morphisms $d_1\to d_3$ and $d_2\to d_3$.
• For any two parallel morphisms $f,g:d_1\to d_2$ in $D$, there exists a morphism $h:d_2\to d_3$ such that $h f = h g$.

One may think of $(d \stackrel{f}{\to} d') \mapsto (F(d) \stackrel{F(f)}{\to} F(d')$ as witnessing that $F(d)$ is “smaller than” or “sitting inside” $F(d')$ in some sense, and $colim F$ is then the “largest” of all these objects, the limiting object.

# Ind-Objects

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, 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

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

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 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 two 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 finitely presentable 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}

So then one defines

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

### as filtered colimits in presheaves

Another, equivalent, definition is to let $ind$-$C$ be the full subcategory of the presheaf category $[C^{op},Set]$ determined by those functors which are filtered colimits of representables. This is reasonable since $[C^{op},Set]$ is the free cocompletion of $C$, so $ind$-$C$ defined in this way is its “free cocompletion under filtered colimits.”

### 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 catgeory Grp of groups is the ind-category of the category of finitely generated groups.

• The catgeory 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.

Last revised on May 29, 2012 at 22:04:00. See the history of this page for a list of all contributions to it.