Fibred 2-categories and bicategories

TL;DR

This paper extends the concept of fibred categories to fibred 2-categories and bicategories, providing a generalized framework with new constructions and stability properties for these higher categorical structures.

Contribution

It introduces fibred 2-categories and bicategories, describes their Grothendieck constructions, and explores their stability and composition properties.

Findings

Fibred 2-categories correspond to 2-functors into 2-Cat.

Fibred bicategories correspond to trihomomorphisms into Bicat.

Fibrations are closed under composition and stable under equiv-comma.

Abstract

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms from a bicategory into Bicat. We describe the Grothendieck construction for each kind of fibration and present a few examples of each. Fibrations in our sense, between bicategories, are closed under composition and are stable under equiv-comma. The free such fibration on a homomorphism is obtained by taking an oplax comma along an identity.