nLab
2-functor

Contents

Contents

Idea

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 ‘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.

Definition

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

Let \mathfrak{C} and 𝔇\mathfrak{D} be 2-categories. A pseudofunctor F:𝔇F:\mathfrak{C}\to\mathfrak{D} consists of

  • A function P:Ob Ob 𝔇P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}, and for each pair of objects A,BOb A,B\in Ob_\mathfrak{C} a functor
P A,B:(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 pair of horizontally composable 1-cells (f,g)Ob (B,C)×Ob (A,B)(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}, a 22-cell isomorphism γ f,g:P(gf)P(g)P(f)\gamma_{f,g}:P(g\circ f)\Rightarrow P(g)\circ P(f) called the associator as below
  • For each object object AOb A\in Ob_\mathfrak{C}, a 22-cell isomorphism ι A:P(1 A)1 P(A)\iota_A:P(1_A)\Rightarrow1_{P(A)} called the unitor as below

These are subject to the following two axioms:

  1. For any composable triplet of 11-cells (f,g,h)Ob (C,D)×Ob (B,C)×Ob (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
γ fg,h(γ f,g1 P(h))=γ f,gh(1 P(f)γ g,h),\gamma_{f\circ g,h}\circ(\gamma_{f,g}\star 1_{P(h)})=\gamma_{f,g\circ h}\circ(1_{P(f)}\star\gamma_{g,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 (B,C)×Ob (A,B)(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)} we have that
ι B1 P(g)=γ 1 B,g,\iota_B\star 1_{P(g)}=\gamma_{1_B,g},
1 P(f)ι B=γ f,1 B,1_{P(f)}\star\iota_B=\gamma_{f,1_B},

as illustrated by the commutative 22-cell diagrams below

           

Lax Functor

To obtain the notion of a lax functor we only require that the associators γ f,g\gamma_{f,g} and unitors ι A\iota_A 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.

Strict 2-Functor

To obtain the notion of a strict 22-functor we require that the associators γ f,g\gamma_{f,g} and unitors ι A\iota_A be 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.

Discussion

There is a notion of a ‘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 ‘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 “22-functor” and nn-functor in general, see higher functor.)


  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 July 29, 2019 at 03:27:11. See the history of this page for a list of all contributions to it.