# Filtered category

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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 $J^\left\{mathrm\left\{op$ 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.

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