# The Azimuth Project Fibred 2-categories and bicategories (Rev #3)

## Idea

The paper is essentially broken into three parts. In the first part, he gives a Grothendieck construction for 2-categories. In the second part, he gives a Grothendieck construction for bicategories. In the third part, he investigates how fibrations of bicategories behave under composition, pullback, and comma.

## Fibred 2-categories

### Definitions

Definition. A 2-functor $P \colon E \to B$ is a 2-fibration if

1. for any $e \in E$ and $f \colon b \to Pe$, there is a cartesian 1-cell $h \colon a \to e$ with $Ph = f$;
2. for any $g \in E$ and $\alpha \colon f \Rightarrow Pg$, there is a cartesian 2-cell $\sigma \colon f \Rightarrow g$ with $P \sigma = \alpha$;
3. the horizontal composite of any two cartesian 2-cells is cartesian.

### Equivalence

Theorem. The Grothendieck construction is the action on objects of a 3-functor

$el \colon [B^{coop}, 2\mathsf{Cat}] \to 2\mathsf{Fib}_s(B)$

and this is an equivalence.