Exponentiable virtual double categories and presheaves for double categories

TL;DR

This paper develops a theory of exponentiable virtual double categories, showing that the 2-category of pseudo double categories and lax functors can be enriched in virtual double categories, simplifying complex proofs and aspects of Yoneda theory.

Contribution

It introduces a universal property of lax functor categories via exponentiability in virtual double categories, providing a new perspective that simplifies existing proofs and theory.

Findings

Pseudo double categories are exponentiable as virtual double categories.

The 2-category of pseudo double categories is enriched in virtual double categories.

Simplifies aspects of Yoneda theory for pseudo double categories.

Abstract

Given a pair of pseudo double categories $\mathbb A$ and $\mathbb B$, the lax functors from $\mathbb A$ to $\mathbb B$, along with their transformations, modules, and multimodulations, assemble into a virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$. We exhibit a universal property of this construction by observing that it arises naturally from the consideration of exponentiability for virtual double categories. In particular, we show that every pseudo double category is exponentiable as a virtual double category, whereby the virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$ of lax functors arises as the virtual double category $\mathbf{\mathbb Mod}(\mathbb B^{\mathbb A})$ of monads and modules in the exponential $\mathbb B^{\mathbb A}$. We explore some consequences of this characterisation, demonstrating that it facilitates simple proofs of statements that heretofore required unwieldy computations. For instance, we deduce that the 2-category of pseudo double categories and lax functors is enriched in the 2-category of normal virtual double categories, and demonstrate that several aspects of the Yoneda theory of pseudo double categories - such as the correspondence between presheaves and discrete fibrations - are substantially simplified by this perspective.