Interaction Laws of Monads and Comonads
Reykjavik University - Tallinn University of Technology
Abstract:
We introduce and study functor-functor and monad-comonad interaction
laws as mathematical objects to describe interaction of effectful
computations with behaviors of effect-performing machines.
Monad-comonad interaction laws are monoid objects of the monoidal
category of functor-functor interaction laws. We show that, for
suitable generalizations of the concepts of dual and Sweedler dual, the
greatest functor resp. monad interacting with a given functor or
comonad is its dual while the greatest comonad interacting with a given
monad is its Sweedler dual. We relate monad-comonad interaction laws to
stateful runners. We show that functor-functor interaction laws are Chu
spaces over the category of endofunctors taken with the Day convolution
monoidal structure. Hasegawa's glueing endows the category of these Chu
spaces with a monoidal structure whose monoid objects are monad-comonad
interaction laws.
This is joint work with Shin-ya Katsumata (NII, Tokyo) and Exequiel
Rivas (Inria, Paris).