$2$-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 $2$-posets by themselves but instead as special cases of $2$-categories, such as the locally posetal ones.

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

Definition

Explicit definition

A 2-poset is a category $C$ such that

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

$C$ is only a 2-proset if $C$ only satisfies 1-3.

From infinity-categories

Fix a meaning of $\infty$-category, however weak or strict you wish. Then a $2$-poset is an $\infty$-category such that all parallel pairs of $j$-morphisms are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $2$-morphisms, not even whether two given parallel $2$-morphisms are equivalent. This definition may give a concept more general than a locally posetal $2$-category for your preferred definition of $2$-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 $2$-category is the $2$-category Cat of categories, so the motivating example of a $2$-poset is the $2$-poset Pos of posets.

Related concepts

• dagger 2-poset

• n-poset

  • poset

  • 2-poset