(Co)filtered Category

Definition

A category $\mathcal{J}$ is filtered when every finite diagram in $\mathcal{J}$ has a cocone. Dually, $\mathcal{J}$ is cofiltered when every finite diagram in $\mathcal{J}$ has a cone.

Examples

Directed Posets

A directed poset is a non-empty poset in which any two elements have a common upper bound.