nLab
equivalence of 2-categories

Context

2-Category theory

Equality and Equivalence

Contents

Definition

An equivalence of 22-categories is the appropriate notion of equivalence between 2-categories. This consists of:

  • 2-functorsF:CD{}\;F\colon C\to D and G:DCG\colon D\to C, and
  • pseudonatural transformationsGFId C{}\;G \circ F \to Id_C and FGId DF \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 FF is strict, and in the case that it is weak. Note however that in this case GG 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 22-category. There is a stricter notion of equivalence for strict 22-categories, which traditionally is called just a 22-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 22-category, the notion of equivalence for 22-categories can be internalized in any 33-category in a straightforward way. The version above for 22-categories then results from specializing this general definition to the (weak) 33-category 2Cat2 Cat of 22-categories, (weak) 22-functors, pseudonatural transformations, and modifications.

There is one warning to keep in mind here. Every 33-category is equivalent to a semi-strict sort of 33-category called a Gray-category, since it is a category enriched over the monoidal category Gray of strict 22-categories and strict 22-functors. Of course GrayGray itself is a Gray-category, but as such it is not equivalent to the weak 33-category 2Cat2 Cat of weak 22-categories and weak 22-functors.

In particular, an “internal (bi)equivalence” in GrayGray consists of strict 22-functors F,GF,G together with pseudonatural equivalences relating GFG F and FGF G to identities. This is a semistrict notion of equivalence, intermediate between the fully weak notion and the fully strict one.

basic properties of…

References

Last revised on September 16, 2020 at 20:52:57. See the history of this page for a list of all contributions to it.