nLab 3-category

Idea

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.

Definition

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.

Specific versions

• globular strict 3-category

• Gray-category

• tricategory

• 3-groupoid

Examples

• 3-category of fermionic conformal nets (2-Clifford algebras)

Related concepts

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category