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

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.

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 $f:kto\; j$ 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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Last updated on Monday August 06, 2007 at 11:46:20 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Monday August 06, 2007 at 11:46:20 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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