- Filtered category
In

category theory ,**filtered categories**generalize the notion ofdirected 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 acolimit of afunctor $F:J\; o\; C$ where $J$ is a filtered category.**Cofiltered categories**There is a dual notion of

**cofiltered**category. A category $J$ is cofiltered if theopposite 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 afunctor $F:J\; o\; C$ where $J$ is a cofiltered category.

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2010.*

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