Categorical notions of fibration

Abstract

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op}⇝\mathbf{Cat}$, while the special case of discrete fibrations encode presheaves $B^{op}\to \mathbf{Set}$. A two-sided discrete variation encodes functors $B^{op}\times A\to \mathbf{Set}$, 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 $\mathbf{Cat}$, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model $\mathcal{V}$-profunctors internally to $\mathcal{V}$-$\mathbf{Cat}$.

These notes were initially written by the second-named author to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago. A few years later, the now first-named author joined to expand and improve in minor ways the exposition. To appear on "Expositiones Mathematicae"