2-poset

- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- 2-category, (2,1)-category
- 1-category
- 0-category
- (?1)-category?
- (?2)-category?

- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- Kan complex
- quasi-category
- simplicial model for weak ∞-categories?

- algebraic definition of higher category
- stable homotopy theory

A **2-poset** is any of several concepts that generalize 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.

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.

Revised on June 3, 2011 17:28:31
by Mike Shulman
(169.228.177.52)