Definitions

# 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 $f:jto k$ and $f\text{'}:j\text{'}to k$ in $J$,
• for every two parallel arrows $u,v:ito j$ in $J$, there exists an object $k$ and an arrow $w:jto k$ such that $wu=wv$.

A filtered colimit is a colimit of a functor $F:Jto C$ 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 $J^\left\{mathrm\left\{op\right\}\right\}$ 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 $f:kto j$ and $f\text{'}:k to j\text{'}$ in $J$,
• for every two parallel arrows $u,v:jto i$ in $J$, there exists an object $k$ and an arrow $w:kto j$ such that $uw=vw$.

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

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