nLab 2-functor




A 22-functor is the categorification of the notion of a functor to the setting of 2-categories. At the 2-categorical level there are several possible versions of this notion one might want depending on the given setting, some of which collapse to the standard definition of a functor between categories when considered on 22-categories with discrete hom-categories (viewed as 11-categories). The least restrictive of these is a lax functor, and the strictest is (appropriately) called a strict 2-functor.

For the various separate definitions that do collapse to standard functors, see:

There is also a notion of β€˜lax functor’, however this notion does not necessarily yield a standard functor when considered on discrete hom-categories.

For the generalisation of this to higher categories, see semistrict higher category.


Here we present explicitly the definition for the middling notion of a pseudofunctor, and comment on alterations that yield the stronger and weaker notions.

Pseudofunctor between strict 22-categories

Let β„­\mathfrak{C} and 𝔇\mathfrak{D} be strict 2-categories. A pseudofunctor P:ℭ→𝔇P:\mathfrak{C}\to\mathfrak{D} consists of

  • A function P:Ob β„­β†’Ob 𝔇P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}.

  • For each pair of objects A,B∈Ob β„­A,B\in Ob_\mathfrak{C} a functor

P A,B:β„­(A,B)→𝔇(P(A),P(B)). P_{A,B}:\mathfrak{C}(A,B)\to\mathfrak{D}(P(A),P(B)).

We will generally write the function and functors as PP.

  • For each triplet of objects A,B,C∈Ob β„­A,B,C\in Ob_\mathfrak{C}, a natural isomorphism

whose components are 22-cell isomorphisms Ξ³ f,g:P(f∘g)β‡’P(f)∘P(g)\gamma_{f,g}:P(f\circ g) \Rightarrow P(f)\circ P(g) as below

  • For each object object A∈Ob β„­A\in Ob_\mathfrak{C}, a natural isomorphism

where 11 denotes the terminal category and id Aid_A is the identity-selecting functor at AA. Its component is a 22-cell isomorphism ΞΉ *:P(1 A)β‡’1 P(A)\iota_{_*}:P(1_A)\Rightarrow 1_{P(A)} as below

These are subject to the following axioms:

  1. For any composable triplet of 11-cells (f,g,h)∈Ob β„­(C,D)Γ—Ob β„­(B,C)Γ—Ob β„­(A,B)(f,g,h)\in Ob_{\mathfrak{C}(C,D)}\times Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)} we have that
    (Ξ³ f,g⋆1 P(h))∘γ f∘g,h=(1 P(f)⋆γ g,h)∘γ f,g∘h, (\gamma_{f,g}\star 1_{P(h)})\circ\gamma_{f\circ g,h}=(1_{P(f)}\star\gamma_{g,h})\circ\gamma_{f,g\circ h},

    where ∘\circ denotes vertical composition and ⋆\star denotes horizontal composition, as illustrated by the following commutative 22-cell diagram in 𝔇(P(A),P(D))\mathfrak{D}(P(A),P(D)):

  1. For any composable 11-cells (f,g)∈Ob β„­(B,C)Γ—Ob β„­(A,B)(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)} we have that
    ΞΉ B⋆1 P(g)=Ξ³ 1 B,g βˆ’1, \iota_B\star 1_{P(g)}=\gamma_{1_B,g}^{-1},
    1 P(f)⋆ι B=Ξ³ f,1 B βˆ’1,1_{P(f)}\star\iota_B=\gamma_{f,1_B}^{-1},

    as illustrated by the commutative 22-cell diagrams below

Β  Β  Β  Β  Β  Β 

Lax functor

To obtain the notion of a lax functor we only require that the coherence morphisms Ξ³ f,g βˆ’1\gamma_{f,g}^{-1} and ΞΉ A βˆ’1\iota_A^{-1} be 22-cells, not necessarily 22-cell isomorphisms. This prevents us from going back and forth between preimages and images of identity 11-cells and horizontally composed 11-cells/22-cells. Similarly, to obtain an oplax functor we reverse the direction of these 2-cells.

Strict 2-Functor

To obtain the notion of a strict 22-functor we require that Ξ³ f,g\gamma_{f,g} and ΞΉ A\iota_A are identity arrows, so horizontal composition and 11-cell identities literally factor through each functor in the same way vertical composition and 22-cell identities do.


There is a notion of a β€˜weak 2-category’, however it usually doesn't make sense to speak of strict 22-functors between weak 22-categories1, but it does make sense to speak of lax (or β€˜weak’) 22-functors between strict 22-categories. Indeed, the weak 33-category Bicat of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to its full sub-3-category spanned by the strict 2-categories. However, it is not equivalent to the 33-category Str2Cat? of strict 22-categories, strict 22-functors, transformations, and modifications. (For discussion of the terminological choice β€œ22-functor” and nn-functor in general, see higher functor.)



(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.

basic properties of…


Textbook accounts:

  1. Although there are certain contexts in which it does. For instance, there is a model structure on the category of bicategories and strict 2-functors between them, which models the homotopy theory of bicategories and weak 2-functors. ↩

Last revised on April 20, 2024 at 07:39:09. See the history of this page for a list of all contributions to it.