nLab equivalence of 2-categories

Idea

The notion of equivalence of $2$-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 $C$ and $D$ consists of

1. 2-functors$\;$ $F \,\colon\, \mathcal{C} \to \mathcal{D}$ and $G \,\colon\, \mathcal{D} \to \mathcal{C}$,

2. pseudonatural transformations${}\;G \circ F \to Id_{\mathcal{C}}$ and $F \circ G \to Id_{\mathcal{D}}$ which are themselves equivalences,

Remark

Def. 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 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 $2$-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.

Properties

Proposition

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

1. essentially surjective:

   surjective on equivalence classes of objects: $\pi_0(F) \;\colon\; \pi_0(\mathcal{C}) \twoheadrightarrow \pi_0(\mathcal{D})\;$,

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

   for each pair of objects $X,\, Y \in \mathcal{C}$ the component functor is an equivalence of hom-categories $F_{X,Y} \,\colon\, \mathcal{C}(X,Y) \xrightarrow{\simeq} \mathcal{D}\big(F(X), F(Y)\big)$,

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

   1. essentially full on 1-cells:

      namely that each component functor $F_{X,Y}$ is an essentially surjective functor;

   2. fully faithful on 2-cells:

      namely that each component functor $F_{X,Y}$ 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 $2$-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

  • injective-on-objects functor

  • essentially injective functor

  • embedding of categories

  • essentially surjective functor

    • surjective-on-objects functor

    • bijective-on-objects functor

    • dominant functor

  • split essentially surjective functor

  • essentially surjective and full functor

  • equivalence of categories

  • conservative functor

  • localization functor

  • pseudomonic functor

  • (co)continuous functor

  • final functor, dense functor

• 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

  • homotopy final functor, final (∞,1)-functor

References

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

• Ofer Gabber, Lorenzo Ramero, Def. 2.49 with Cor. 2.4.30 in: Foundations for almost ring theory (arXiv:math/0409584)

• Weizhe Zheng, Def. 1.6 - Lem. 1.8 of: Gluing pseudofunctors via $n$-fold categories, J. Homotopy Relat. Struct. 12 189–271 (2017) (arXiv:1211.1877, doi:10.1007/s40062-016-0126-2)

• Niles Johnson, Donald Yau, Section 7 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)