Exponentiable Virtual Double Categories and Presheaves for 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 \(\mathbb {L}{\textbf {ax}} (\mathbb {A}, \mathbb {B})\). 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 \(\mathbb {L}{\textbf {ax}} (\mathbb {A}, \mathbb {B})\) of lax functors arises as the virtual double category \(\mathbb {M}{\textbf {od}} (\mathbb {B}^\mathbb {A})\) of monads and modules in the exponential \(\mathbb {B}^\mathbb {A}\). We explore some consequences of this characterization, demonstrating that it leads to 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.