Monad-comonad interaction laws are a means for specifying communication protocols between effectful computations and coeffectful environments in the paradigm where notions of effectful computation are modelled by monads and notions of coeffectful environment by comonads.
In this talk, I will demonstrate that monad-comonad interaction laws are an instance of measuring maps from López Franco and Vasilakopoulou's Sweedler theory for duoidal categories. The final interacting comonad for a monad and a residual monad arises as the Sweedler hom and the initial residual monad for a monad and an interacting comonad as the Sweedler copower.
I will explain a (co)algebraic characterization of monad-comonad interaction laws and how it leads to descriptions of the Sweedler hom and the Sweedler copower in terms of their coalgebras resp. algebras.
Joint work with Dylan McDermott and Exequiel Rivas.