In category theory, filtered categories generalize the notion of directed 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 $f:j\backslash to\; k$ and $f\text{'}:j\text{'}\backslash to\; k$ in $J$,
• for every two parallel arrows $u$ $u,v:i\backslash to\; j$ in $J$, there exists an object $k$ and an arrow $w:j\backslash to\; k$ such that $wu=wv$.

A filtered colimit is a colimit of a functor $F:J\backslash to\; C$ where $J$ is a filtered category.

Category theory