Filtered category

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 $j\text{'}$ in $J$ there exists an object $k$ and two arrows $f:j\; o\; k$ and $f\text{'}:j\text{'}\; o\; k$ in $J$,
* for every two parallel arrows $u,v:i\; o\; j$ in $J$, there exists an object $k$ and an arrow $w:j\; o\; k$ such that $wu=wv$.

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

Cofiltered categories

There is a dual notion of cofiltered category. 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 $j\text{'}$ in $J$ there exists an object $k$ and two arrows $f:k\; o\; j$ and $f\text{'}:k\; o\; j\text{'}$ in $J$,
* for every two parallel arrows $u,v:j\; o\; i$ in $J$, there exists an object $k$ and an arrow $w:k\; o\; j$ such that $uw=vw$.

A (co)filtered limit is a limit of a functor $F:J\; o\; C$ where $J$ is a cofiltered category.