Interaction morphisms

Tarmo Uustalu, Tallinn U of Technology

I will propose interaction morphisms as a means to specify how an effectful computation is to be run, provided a state in a context. An interaction morphism of a monad T and a comonad D is a family of maps T X x D Y → X x Y natural in X and Y and subject to some equations. Interaction morphisms turn out to be a natural concept with a number of neat properties. In particular, interaction morphisms are the same as monoids in a certain monoidal category; interaction morphisms of T and D are in a bijective correspondence with carrier-preserving functors between the categories of coalgebras of D and stateful runners of T (monad morphisms from T to state monads); they are also in a bijective correspondence with monad morphisms from T to a monad induced in a certain way by D.

This is joint work with Shin-ya Katsumata (University of Kyoto).