2-poset in nLab

-posets

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), there is a binary relation ≤ A,B\leq_{A, B} on Hom(A,B)Hom(A, B)
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 j≥2j \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

• Pos

• Rel

• simplex category

• Lat

• DistLat

• Frm

• Locale

• HeytAlg

• BoolAlg

• allegory

• bicategory of relations

• first-order hyperdoctrine with equality

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.

• dagger 2-poset

• n-poset

  • poset

  • 2-poset