# Contents

## Definition

A $2$-category $C$ is locally posetal or locally partially ordered or Pos-enriched if every hom-category $C\left(x,y\right)$ 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); 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 $\mathrm{Pos}$.

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 $\mathrm{Pos}$. 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

Revised on August 26, 2012 23:52:07 by Rod Mc Guire? (68.162.156.188)