filtered category

The cardinality of a small category is the cardinality of its set of arrows.

A category $C$ is $\kappa$-filtered where $\kappa$ is an infinite regular cardinal if for every diagram in $d: D\to C$ of cardinality smaller than $\kappa$ there is a cone over $d$.

A category is filtered if it is $\omega$-filtered. Alternatively it is a nonempty category such that certain elementary types of finite diagrams have cones.

The importance is in the fact that if $C$ is small and filtered, the colimits of functors $C\to Set$ commute with finite limits in $Set$, and conversely, a small category is filtered iff the colimits of all $C$-colimits in $Set$ commute with finite limits.

See filtered limit in $n$Lab.

Created on February 2, 2011 at 20:41:22. See the history of this page for a list of all contributions to it.