definition. filtered category [rosiak2022sheaf, def. 285] [tt-004Z]

✍️source

A category \({\cal J}\) is filtered (or directed) if

• it is not empty
• for every pair of objects \(J, J^{\prime } \in {\cal J}\), there exists \(K \in {\cal J}\) and \(f: J \to K\) and \(f': J' \to K\)
• for every two parallel arrows \(u, v: i \rightarrow j\) in \({\cal J}\), there exists \(K \in {\cal J}\) and \(w: J \to K\) such that \[u \mathbin {\bullet } w = v \mathbin {\bullet } w\]

This is a generalization of the notion of a (upward) directed poset from order theory.