Higher category theory

higher category theory

Basic concepts

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Universal constructions

Extra properties and structure

1-categorical presentations


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.


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.


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 3, 2011 at 17:28:31. See the history of this page for a list of all contributions to it.