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:EBP \colon E \to B is a 2-fibration if

  1. for any eEe \in E and f:bPef \colon b \to Pe, there is a cartesian 1-cell h:aeh \colon a \to e with Ph=fPh = f;
  2. for any gEg \in E and α:fPg\alpha \colon f \Rightarrow Pg, there is a cartesian 2-cell σ:fg\sigma \colon f \Rightarrow g with Pσ=αP \sigma = \alpha;
  3. the horizontal composite of any two cartesian 2-cells is cartesian.

Construction

Equivalence

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

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

and this is an equivalence.

Fibred bicategories