nLab 2-poset

-posets

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 (categorify) the notion of posets one step in higher category theory. One does not usually hear about 22-posets by themselves but instead as special cases of 2 2 -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 n n -posets.

Definition

Explicit definition

A 2-poset is a category CC such that

  1. For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), there is a binary relation R A,BSR \leq_{A, B} S
  2. For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), R A,BRR \leq_{A, B} R.
  3. For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), T:Hom(A,B)T:Hom(A, B), R A,BSR \leq_{A, B} S and S A,BTS \leq_{A, B} T implies R A,BTR \leq_{A, B} T.
  4. For each object A:Ob(C)A:Ob(C) and B:Ob(C)B:Ob(C) and morphism R:Hom(A,B)R:Hom(A, B), S:Hom(A,B)S:Hom(A, B), R A,BSR \leq_{A, B} S and S A,BRS \leq_{A, B} R implies R=SR = S.

CC is only a 2-proset if CC only satisfies 1-3.

From infinity-categories

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.

Last revised on June 7, 2022 at 16:36:56. See the history of this page for a list of all contributions to it.