nLab
equivalence of 2-categories

Definition

An equivalence of $2$-categories is the appropriate notion of equivalence between 2-categories. As used on the nLab, where all n-categories are usually by default “weak,” this consists of:

• (Weak) 2-functors (aka pseudofunctors) $F:C\to D$ and $G:D\to C$, and
• pseudonatural transformations $G\circ F\to {\mathrm{Id}}_C$ and $F\circ G\to {\mathrm{Id}}_D$ which are themselves equivalences, i.e. there are pseudonatural transformations forming their inverses up to isomorphism.

In the literature this sort of equivalence is often called a biequivalence, as it has traditionally been associated with bicategories, the standard algebraic definition of weak $2$-category. There is a stricter notion of equivalence for strict $2$-categories, which traditionally is called just a $2$-equivalence and which on the nLab is called a strict 2-equivalence.

A 2-functor can be made into part of an equivalence iff it is essentially surjective on objects, essentially full on 1-cells, and fully faithful on 2-cells.

Internalization

Just as the notion of equivalence of categories can be internalized in any $2$-category, the notion of equivalence for $2$-categories can be internalized in any $3$-category in a straightforward way. The version above for $2$-categories then results from specializing this general definition to the (weak) $3$-category $2\mathrm{Cat}$ of $2$-categories, (weak) $2$-functors, pseudonatural transformations, and modifications.

There is one warning to keep in mind here. Every $3$-category is equivalent to a semi-strict sort of $3$-category called a Gray-category, since it is a category enriched over the monoidal category Gray of strict $2$-categories and strict $2$-functors. Of course $\mathrm{Gray}$ itself is a Gray-category, but as such it is not equivalent to the weak $3$-category $2\mathrm{Cat}$ of weak $2$-categories and weak $2$-functors.

In particular, an “internal (bi)equivalence” in $\mathrm{Gray}$ consists of strict $2$-functors $F,G$ together with pseudonatural equivalences relating $GF$ and $FG$ to identities. This is a semistrict notion of equivalence, intermediate between the fully weak notion and the fully strict one.

Related concepts

• equality, isomorphism, equivalence

• weak equivalence, homotopy equivalence, weak homotopy equivalence

• homotopy equivalence of toposes

• equivalence in an (∞,1)-category

• equivalence of categories, adjoint functor

• equivalence of 2-categories, 2-adjunction

• equivalence of (∞,1)-categories, adjoint (∞,1)-functor