nLab
equivalence of 2-categories

Context

2-Category theory

Equality and Equivalence

Contents

Idea

The notion of equivalence of 22-categories is the appropriate notion of equivalence between 2-categories, categorifying the notion of equivalence of categories: a pair of 2-functors back and forth between 2-categories, which are inverse to each other, up to pseudonatural equivalence.

Definition

Definition

An equivalence between 2-categories CC and DD consists of

  1. 2-functors\; F:π’žβ†’π’ŸF \,\colon\, \mathcal{C} \to \mathcal{D} and G:π’Ÿβ†’π’žG \,\colon\, \mathcal{D} \to \mathcal{C},

  2. pseudonatural transformationsG∘Fβ†’Id π’ž{}\;G \circ F \to Id_{\mathcal{C}} and F∘Gβ†’Id π’ŸF \circ G \to Id_{\mathcal{D}} which are themselves equivalences,

Remark

Def. 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 Lack 2002, Ex, 3.1.

Remark

In the literature this sort of equivalence in Def. 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.

Properties

Proposition

(recognition of equivalences of 2-categories assuming the axiom of choice)
Assuming the axiom of choice, a 2-functor F:π’žβ†’π’ŸF \,\colon\, \mathcal{C} \xrightarrow{\;} \mathcal{D} is an equivalence of 2-categories precisely if it is

  1. essentially surjective:

    surjective on equivalence classes of objects: Ο€ 0(F):Ο€ 0(π’ž)β† Ο€ 0(π’Ÿ)\pi_0(F) \;\colon\; \pi_0(\mathcal{C}) \twoheadrightarrow \pi_0(\mathcal{D})\;,

  2. fully faithful (e.g. Gabber & Ramero 2004, Def. 2.4.9 (ii)):

    for each pair of objects X,Yβˆˆπ’žX,\, Y \in \mathcal{C} the component functor is an equivalence of hom-categories F X,Y:π’ž(X,Y)β†’β‰ƒπ’Ÿ(F(X),F(Y))F_{X,Y} \,\colon\, \mathcal{C}(X,Y) \xrightarrow{\simeq} \mathcal{D}\big(F(X), F(Y)\big),

    which by the analogous theorem for 1-functors (this Prop.) means equivalently that FF is (e.g. Johnson & Yau 2020, Def. 7.0.1)

    1. essentially full on 1-cells:

      namely that each component functor F X,YF_{X,Y} is an essentially surjective functor;

    2. fully faithful on 2-cells:

      namely that each component functor F X,YF_{X,Y} is a fully faithful functor.

This is classical folklore. It is made explicit in, e.g. Gabber & Ramero 2004, Cor. 2.4.30; Johnson & Yau 2020, Thm. 7.4.1.

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 β€œinternal (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 28, 2021 at 04:54:39. See the history of this page for a list of all contributions to it.