---- > [!definition] Definition. ([[filtered category]]) > > A nonempty [[category]] $\mathsf{I}$ is **filtered** if: > 1. For each pair $x,y$ of objects, there is an object $z$ and morphisms $x \to z$ and $y \to z$ > 2. For each pair of parallel morphisms $u:x \to y$ and $v: x \to y$, there exists a morphism $w: y \to z$ such that $w \circ u=w \circ v$. > > From Vakil: > ![[CleanShot 2025-01-30 at [email protected]]] > > [!justification] > If $\mathsf{I}$ is a [[poset]], then one recovers the notion of a [[filtered poset|filtered poset/directed set]]. In particular, condition (2) dissolves because parallel morphisms don't exist in [[poset|posets]]. ^justification ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```