nLab 2-functor

Idea

A $2$-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 $2$-categories with discrete hom-categories (viewed as $1$-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:

• strict 2-functor
• pseudo functor

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 between strict $2$-categories

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

• A function $P:Ob_\mathfrak{C}\to Ob_\mathfrak{D}$.

• For each pair of objects $A,B\in Ob_\mathfrak{C}$ a functor

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

We will generally write the function and functors as $P$.

• For each triplet of objects $A,B,C\in Ob_\mathfrak{C}$, a natural isomorphism

whose components are $2$-cell isomorphisms $\gamma_{f,g}:P(f\circ g) \Rightarrow P(f)\circ P(g)$ as below

• For each object object $A\in Ob_\mathfrak{C}$, a natural isomorphism

where $1$ denotes the terminal category and $id_A$ is the identity-selecting functor at $A$. Its component is a $2$-cell isomorphism $\iota_{_*}:P(1_A)\Rightarrow 1_{P(A)}$ as below

These are subject to the following axioms:

1. For any composable triplet of $1$-cells $(f,g,h)\in Ob_{\mathfrak{C}(C,D)}\times Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
   $$(\gamma_{f,g}\star 1_{P(h)})\circ\gamma_{f\circ g,h}=(1_{P(f)}\star\gamma_{g,h})\circ\gamma_{f,g\circ h},$$

   where $\circ$ denotes vertical composition and $\star$ denotes horizontal composition, as illustrated by the following commutative $2$-cell diagram in $\mathfrak{D}(P(A),P(D))$:

1. For any composable $1$-cells $(f,g)\in Ob_{\mathfrak{C}(B,C)}\times Ob_{\mathfrak{C}(A,B)}$ we have that
   $$\iota_B\star 1_{P(g)}=\gamma_{1_B,g}^{-1},$$
   $$1_{P(f)}\star\iota_B=\gamma_{f,1_B}^{-1},$$

   as illustrated by the commutative $2$-cell diagrams below

           

Lax 2-Functor

To obtain the notion of a lax functor we only require that the coherence morphisms $\gamma_{f,g}$ and $\iota_A$ be $2$-cells, not necessarily $2$-cell isomorphisms. This prevents us from going back and forth between preimages and images of identity $1$-cells and horizontally composed $1$-cells/$2$-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 $2$-functor we require that $\gamma_{f,g}$ and $\iota_A$ are identity arrows, so horizontal composition and $1$-cell identities literally factor through each functor in the same way vertical composition and $2$-cell identities do.

Remark

There is a notion of a ‘weak 2-category’, however it usually doesn't make sense to speak of strict $2$-functors between weak $2$-categories¹, but it does make sense to speak of lax (or ‘weak’) $2$-functors between strict $2$-categories. Indeed, the weak $3$-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 $3$-category Str2Cat? of strict $2$-categories, strict $2$-functors, transformations, and modifications. (For discussion of the terminological choice “$2$-functor” and $n$-functor in general, see higher functor.)

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.

Related concepts

• function

• functor

• 2-functor / pseudofunctor / (2,1)-functor

• n-functor

• (∞,1)-functor

• (∞,n)-functor

basic properties of…

• functors

  • faithful functor

  • full functor

  • full and faithful functor

  • injective-on-objects functor

  • essentially injective functor

  • 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

  • pseudomonic functor

  • (co)continuous functor

  • final 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

Textbook accounts:

• Ofer Gabber, Lorenzo Ramero, Def. 2.1.14 in: Foundations for almost ring theory (arXiv:math/0409584)

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

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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. ↩