Filtered category

From Wikipedia, the free encyclopedia

In category theory, filtered categories generalize the notion of directed set.

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 $w u = w v$ .

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

[edit] Cofiltered categories

There is a dual notion of cofiltered category. A category $J$ is cofiltered if the opposite category $J op$ 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 $u w = v w$ .