Filtered category

In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category which will be recalled below.

A category ${\displaystyle J}$ is filtered when

A filtered colimit is a colimit of a functor ${\displaystyle F:J\to C}$ where ${\displaystyle J}$ is a filtered category.

A category ${\displaystyle J}$ is cofiltered if the opposite category ${\displaystyle J^{\mathrm {op} }}$ is filtered. In detail, a category is cofiltered when

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