Dinaturality for Double Categories

Abstract

In this paper we extend the concept of dinaturality to the setting of double categories. We introduce the dinatural versions of double-categorical transformations and modifications, and show that ordinary natural transformations and modifications correspond to dinatural ones between dummy functors. Although dinatural transformations don't generally compose with each other, they do compose with natural transformations, and we investigate the algebra of this composition. In our motivating example of dinaturality for double categories, we derive the caps and cups of Eilenberg-Kelly graphs for extranatural transformations as dinatural transformation components, and the corresponding adjunction laws as (di)modification components. In an appendix we extend the surface diagram calculus for the locally cubical Gray category of small double categories to include dinatural constructions.