A Characterization of Pro-representable Virtual Double Categories

Abstract

Virtual double categories have been shown to provide an effective framework for formal category theory, from characterizations of adjoints and liftings to descriptions of pointwise Kan extensions and Yoneda structures. In studying virtual double categories themselves, an interesting question comes from asking what kind of structure virtual double categories are enriched over. In this talk, we take a first step in answering this question by characterizing the exponentiable, or pro-representable, virtual double categories. We will provide examples of pro-representable virtual double categories, including pseudo double categories and cospan virtual double categories, while also providing a simple counter-example to the claim that all virtual double categories are exponentiable. If there is time, we will discuss another internal-hom candidate coming from Libkind, Myers, Carlson, and Brown’s theory of loose bimodules.