nLab locally posetal 2-category

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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), or a locally thin 2-category, where every hom-category is a thin category; 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

Last revised on July 10, 2024 at 14:51:33. See the history of this page for a list of all contributions to it.