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

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

A $3$-category is any of several concepts that generalize $2$-categories one step in higher category theory. The original notion is that of a globular strict 3-category, but the one most often used here is that of a tricategory. The concept generalizes to $n$-categories.

Fix a meaning of $\infty$-category, however weak or strict you wish. Then a **$3$-category** is an $\infty$-category such that every 4-morphism is an equivalence, and all parallel pairs of $j$-morphisms are equivalent for $j \geq 4$. Thus, up to equivalence, there is no point in mentioning anything beyond $3$-morphisms, except whether two given parallel $3$-morphisms are equivalent. This definition may give a concept more general than your preferred definition of $3$-category, but it will be equivalent; basically, you may have to rephrase equivalence of $3$-morphisms as equality.

Last revised on May 7, 2013 at 23:48:40. See the history of this page for a list of all contributions to it.