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.
A 2-poset is a category $C$ such that
$C$ is only a 2-proset if $C$ only satisfies 1-3.
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.
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.
2-poset
Last revised on September 19, 2022 at 20:00:32. See the history of this page for a list of all contributions to it.