The pseudomonad perspective on double categories, with applications to double profunctors

Abstract: Just as a category is a monad in the bicategory Span(Set), so a double category is a pseudomonad in the analogous tricategory Span(Cat). Pseudo-bimodules of between these give a notion of double profunctor, and these in turn specialise to recover double functors. Composition of double profunctors uses a two-dimensional colimit which, unlike the reflexive coequalisers of the standard internal category setting, are stable under pullback in Cat. We extend Grandis and Pare’s strictification results for double categories and double functors to analogous strictification results for double profunctors. We recover the tricategory PsDblCat whose morphisms are double functors, and describe work in progress, joint with Nicola Gambino, towards a tricategory PsDblProf whose morphisms are double profunctors. Finally, a dual specialisation of double profunctors will also be described giving a notion of double cofunctor, establishing the tricategory that these maps form, and providing a strictification result for them.

category theory

Audience: researchers in the topic

Second Virtual Workshop on Double Categories