nLab locally posetal 2-category

Definition

A $2$-category $C$ is locally posetal or locally partially ordered or Pos-enriched if every hom-category $C(x,y)$ is a poset - an object of the category Pos of partial orders. One can also consider a locally preordered $2$-category, where every hom-category is a proset (a preordered set), or a locally thin 2-category, where every hom-category is a thin category; up to equivalence of $2$-categories, these aren't any more general.

Locally posetal $2$-categories are the usual model of 2-posets, aka (1,2)-categories. Just as the motivating example of a $2$-category is the $2$-category Cat of categories, so the motivating example of a $2$-poset is the $2$-poset Pos of posets. If you interpret $\Pos$ as a full sub-2-category of $\Cat$, then it is indeed locally posetal. Similarly, the $2$-category of prosets is a locally preordered $2$-category that is equivalent to $Pos$.

Compare the notion of partially ordered category. A locally partially ordered category is a category enriched over the category Pos of posets, while a partially ordered category is a category internal to $Pos$. Similarly, a locally partially ordered category is a special kind of $2$-category, while a partially ordered category is a special kind of double category.

Examples

• Pos
• Rel