# Filtered category

(Redirected from Filtered colimit)

In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category which will be recalled below.

## Filtered categories

A category $J$  is filtered when

• it is not empty,
• for every two objects $j$  and $j'$  in $J$  there exists an object $k$  and two arrows $f:j\to k$  and $f':j'\to k$  in $J$ ,
• for every two parallel arrows $u,v:i\to j$  in $J$ , there exists an object $k$  and an arrow $w:j\to k$  such that $wu=wv$ .

A filtered colimit is a colimit of a functor $F:J\to C$  where $J$  is a filtered category.

## Cofiltered categories

A category $J$  is cofiltered if the opposite category $J^{\mathrm {op} }$  is filtered. In detail, a category is cofiltered when

• it is not empty
• for every two objects $j$  and $j'$  in $J$  there exists an object $k$  and two arrows $f:k\to j$  and $f':k\to j'$  in $J$ ,
• for every two parallel arrows $u,v:j\to i$  in $J$ , there exists an object $k$  and an arrow $w:k\to j$  such that $uw=vw$ .

A cofiltered limit is a limit of a functor $F:J\to C$  where $J$  is a cofiltered category.

## Ind-objects and pro-objects

Given a small category $C$ , a presheaf of sets $C^{op}\to Set$  that is a small filtered colimit of representable presheaves, is called an ind-object of the category $C$ . Ind-objects of a category $C$  form a full subcategory $Ind(C)$  in the category of functors (presheaves) $C^{op}\to Set$ . The category $Pro(C)=Ind(C^{op})^{op}$  of pro-objects in $C$  is the opposite of the category of ind-objects in the opposite category $C^{op}$ .

## κ-filtered categories

There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a cocone over any diagram in $J$  of the form $\{\ \ \}\rightarrow J$ , $\{j\ \ \ j'\}\rightarrow J$ , or $\{i\rightrightarrows j\}\rightarrow J$ . The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for any finite diagram; in other words, a category $J$  is filtered (according to the above definition) if and only if there is a cocone over any finite diagram $d:D\to J$ .

Extending this, given a regular cardinal κ, a category $J$  is defined to be κ-filtered if there is a cocone over every diagram $d$  in $J$  of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)

A κ-filtered (co)limit is a (co)limit of a functor $F:J\to C$  where $J$  is a κ-filtered category.