Categorical notions of fibration

TL;DR

This paper explores various categorical fibrations, including their internal definitions in 2-categories and bicategories, and highlights the unexpected role of dual two-sided cofibrations in modeling al V-profunctors.

Contribution

It extends the internal theory of fibrations to bicategories and reveals the necessity of dual cofibrations for modeling al V-profunctors internally.

Findings

Internal definitions of fibrations in bicategories are possible.

Dual two-sided cofibrations are essential for al V-profunctors.

The work generalizes classical fibrations to higher categorical contexts.

Abstract

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided discrete variation encodes functors $B^{op} \times A \to {\bf Set}$, which are also known as profunctors from $A$ to $B$. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to ${\bf Cat}$, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model $\cal V$-profunctors internally to $\cal V$-$\bf Cat$.