Discrete Double Fibrations

TL;DR

This paper explores the concept of discrete double fibrations, establishing an equivalence with a category of elements construction for presheaves on small double categories, extending classical fibrations to a double categorical setting.

Contribution

It identifies the discrete fibration concept for presheaves on small double categories and proves an equivalence of virtual double categories via the category of elements construction.

Findings

Discretizes the notion of fibrations in double categories.

Establishes an equivalence between presheaves and discrete double fibrations.

Extends classical category theory concepts to double categorical frameworks.

Abstract

Presheaves on a small category are well-known to correspond via a category of elements construction to ordinary discrete fibrations over that same small category. Work of R. Par\'e proposes that presheaves on a small double category are certain lax functors valued in the double category of sets with spans. This paper isolates the discrete fibration concept corresponding to this presheaf notion and shows that the category of elements construction introduced by Par\'e leads to an equivalence of virtual double categories.