Notions of 2-functor

There are three possible generalizations of functor to a bicategory. The perservation of the composition and unit for 1-morphisms must be witnessed by 2-morphisms, but we are faced with the choice of whether these 2-morphisms should be directed or invertible, and if the latter, in which direction.¹

Lax 2-functor
Oplax 2-functor
2-functor

Lax 2-functor

A lax 2-functor $𝐹 : ℭ \to 𝔇$ between bicategories is a generalization of a functor consisting of #m/def/cat/bi

a map $𝐹 0 : ℭ 0 \to 𝔇 0 : 𝐴 \mapsto 𝐹 𝐴$ ;
For every $𝐴, 𝐵 \in ℭ$ , a functor $𝐹 𝐴, 𝐵 : ℭ (𝐴, 𝐵) \to ℭ (𝐹 0 𝐴, 𝐹 0 𝐵)$ ;

Notation

We denote the corresponding map on 1-morphisms by $𝐹 1 : ℭ 1 (𝐴, 𝐵) \to ℭ 1 (𝐹 0 𝐴, 𝐹 0 𝐵)$ , and the corresponding function on 2-morphisms by $𝐹 2 : ℭ 2 (𝐴, 𝐵) (𝑓, 𝑔) \to ℭ 2 (𝐹 0 𝐴, 𝐹 0 𝐵) (𝐹 1 𝑓, 𝐹 1 𝑔)$ .

with directed functorality witnessed by

for any $𝐴, 𝐵, 𝐶 \in ℭ 0$ , a natural transformation called the compositor with components $𝛾 𝑔, 𝑓 : (𝐹 1 𝑔) (𝐹 1 𝑓) \Rightarrow 𝐹 1 (𝑔 𝑓) : 𝐴 \to 𝐶$ for indices $𝑔 \in ℭ 1 (𝐵, 𝐶)$ and $𝑓 \in ℭ 1 (𝐴, 𝐵)$ ;
for any $𝐴 \in ℭ 0$ a 1-morphism $𝜈 𝑓 : 𝟙 𝐹 0 𝐴 \Rightarrow 𝐹 1 𝟙 𝐴 : 𝐴 \to 𝐴$ called the unitor;

and these data are coherent with associativity

and unitality

$c|https://q.uiver.app/#q=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 \begin{tikzcd}[cramped] & {F_1(\mathbb 1_B \circ f)} && {F_1(f \circ \mathbb 1_A)} & \ {F_1 \mathbb 1_B \circ F_1 f} && {F_1f} && {F_1f \circ F_1 \mathbb 1_A} \ & {\mathbb 1_{F_0 B} \circ F_1 f} && {F_1 f \circ \mathbb 1_{F_0 A}} \arrow["{F_2 \lambda_f}", Rightarrow, from=1-2, to=2-3] \arrow["{F_1 \rho_x}"', Rightarrow, from=1-4, to=2-3] \arrow["{\gamma_{\mathbb 1_B, f}}", Rightarrow, from=2-1, to=1-2] \arrow["{\gamma_{f ,\mathbb 1_A}}"', Rightarrow, from=2-5, to=1-4] \arrow["{\nu \circ F_1 f}", Rightarrow, from=3-2, to=2-1] \arrow["{\lambda_{F_1 f}}"', Rightarrow, from=3-2, to=2-3] \arrow["{\rho_{F_1f}}", Rightarrow, from=3-4, to=2-3] \arrow["{F_1 f \circ \nu}"', Rightarrow, from=3-4, to=2-5] \end{tikzcd}$

for $𝐴 𝑓 \to 𝐵 𝑔 \to 𝐶 ℎ \to 𝐷$ in $ℭ$ .

Oplax 2-functor

If we reverse all the 2-cells in the definition of a lax 2-functor we get an oplax 2-functor. #m/def/cat/bi Equivalently, an oplax 2-functor from $ℭ \to 𝔇$ is a lax 2-functor $ℭ c o \to 𝔇 c o$ , where we have invoked the 2-morphism dual.

2-functor

A (proper) 2-functor² is a lax 2-functor such that all compositors and unitors are isomorphisms. #m/def/cat/bi Hence it is also an oplax 2-functor.

#state/tidy | #lang/en | #SemBr

2021. 2-Dimensional categories, §4 ↩
Also called a pseudofunctor or weak 2-functor. ↩

Notions of 2-functor

Lax 2-functor

Oplax 2-functor

2-functor

Footnotes