Exponentiable Virtual Double Categories and Representability of Exponentials

Double categories, which can be viewed as categories with two types of maps that interact through cells, have proved invaluable throughout both pure and applied mathematics for modelling structures with two relational modes. In many applications of interest, such as categories with maps given by functors and profunctors, it is necessary to allow one of the types of maps to only satisfy composition axioms up to coherent isomorphisms, resulting in the notion of pseudo-double categories. However, in these instances many important constructions on pseudo-double categories, such as the internal hom objects of Paré [1], fail to produce pseudo-double categories in general, and instead produce virtual double categories, where now one of the types of morphisms has a priori no composition operation. Nonetheless, virtual double categories have proved to still encode rich mathematical structures, providing an effective framework for formal category theory, from characterizations of adjoints and liftings to descriptions of pointwise Kan extensions and weighted (co)limits [2].

In studying virtual double categories themselves, the importance of functor categories in ordinary category theory motivates us to study when such functor category-type objects exist for virtual double categories. In this talk, we answer this question by providing a number of equivalent descriptions of the virtual double categories $𝔻$ for which the internal hom functor ${(-)}^{𝔻}$ exists, many of which address the prongs of a conjecture posed by Arkor in [3]. We will also show that one of our characterizations readily extends to a description of the virtual double functors which admit dependent products, generalizing the Conduché condition for exponentiable functors [4]. Throughout we will provide examples of exponentiable virtual double categories, including pseudo-double categories and cospan virtual double categories, along with connections to the theory of exponentiable multicategories, which can be seen as a shadow of the higher categorical theory of exponentiable virtual double categories. We will conclude by discussing extensions of Paré’s work on the representability of hom objects for pseudo-double categories [1] to the case of virtual double categories, where the source of the hom object is exponentiable and the target is a weakly representable virtual double category. These extensions will be shown to provide a generalization of Day convolution and Day’s classification of pro-monoidal categories (i.e. exponentiable multicategories) in terms of bi-cocontinuous monoidal structures on copresheaf categories [5].