Zoran Skoda
filtered category

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

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

A category is filtered if it is ω\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 CC is small and filtered, the colimits of functors CSetC\to Set commute with finite limits in SetSet, and conversely, a small category is filtered iff the colimits of all CC-colimits in SetSet commute with finite limits.

See filtered limit in nnLab.

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