nLab
locally posetal 2-category

Contents

Definition

A 22-category CC is locally posetal or locally partially ordered or Pos-enriched if every hom-category C(x,y)C(x,y) is a poset - an object of the category Pos of partial orders. One can also consider a locally preordered 22-category, where every hom-category is a proset (a preordered set); up to equivalence of 22-categories, these aren't any more general.

Locally posetal 22-categories are the usual model of 2-posets, aka (1,2)-categories. Just as the motivating example of a 22-category is the 22-category Cat of categories, so the motivating example of a 22-poset is the 22-poset Pos of posets. If you interpret Pos\Pos as a full sub-2-category of Cat\Cat, then it is indeed locally posetal. Similarly, the 22-category of prosets is a locally preordered 22-category that is equivalent to PosPos.

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 PosPos. Similarly, a locally partially ordered category is a special kind of 22-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)