nLab
equivalence of 2-categories

Context

2-Category theory

Equality and Equivalence

Equivalence of 22-categories

Definition

An equivalence of 22-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:

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 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 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.

Revised on September 26, 2012 14:02:00 by Jamie Vicary (81.135.52.3)