nLab
2-poset

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

22-posets

A 2-poset is any of several concepts that generalize posets one step in higher category theory. One does not usually hear about 22-posets by themselves but instead as special cases of 22-categories, such as the locally posetal ones.

22-posets can also be called (1,2)-categories, being a special case of (n,r)-categories. The concept generalizes to nn-posets.

Definition

Fix a meaning of \infty-category, however weak or strict you wish. Then a 22-poset is an \infty-category such that all parallel pairs of jj-morphisms are equivalent for j2j \geq 2. Thus, up to equivalence, there is no point in mentioning anything beyond 22-morphisms, not even whether two given parallel 22-morphisms are equivalent. This definition may give a concept more general than a locally posetal 22-category for your preferred definition of 22-category, but it will be equivalent if you ignore irrelevant data.

Examples

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.

Revised on June 3, 2011 17:28:31 by Mike Shulman (169.228.177.52)