nLab
equivalence of 2-categories

Definition

An equivalence of $2$-categories is the appropriate notion of equivalence between 2-categories. This consists of:

• 2-functors${}\;F\colon C\to D$ and $G\colon D\to C$, and
• pseudonatural transformations${}\;G \circ F \to Id_C$ and $F \circ G \to Id_D$ which are themselves equivalences, i.e. there are pseudonatural transformations forming their inverses up to isomorphism.

The definition makes sense, and is used, both in the case that $F$ is strict, and in the case that it is weak. Note however that in this case $G$ should be allowed to be weak: see Example 3.1 in Lack2002.

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 (weak or strict) 2-functor can be made into part of an equivalence iff it is essentially surjective on objects, essentially full on 1-cells (i.e. essentially surjective on Hom-categories), 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 $2 Cat$ 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 $Gray$ itself is a Gray-category, but as such it is not equivalent to the weak $3$-category $2 Cat$ of weak $2$-categories and weak $2$-functors.

In particular, an “internal (bi)equivalence” in $Gray$ consists of strict $2$-functors $F,G$ together with pseudonatural equivalences relating $G F$ and $F G$ 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 (2,1)-categories

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

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • essentially surjective functor

    • surjective on objects functor

    • bijective on objects functor

  • essentially surjective and full functor

  • equivalence of categories

• 2-functors

  • full and faithful 2-functor

  • locally fully faithful 2-functor

  • equivalence of 2-categories

• (∞,1)-functors:

  • fully faithful (∞,1)-functor

  • essentially surjective (∞,1)-functor

  • equivalence of (∞,1)-categories

References

• Stephen Lack, A Quillen model structure for 2-categories, K-Theory 26, No. 2, 171-205 (2002). Zentralblatt review author’s webpage

• Weizhe Zheng, prop. 1.6 of Gluing pseudofunctors via $n$-fold categories (arXiv:1211.1877)