Interaction laws of monads and comonads
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.
Keywords
Cite
@article{arxiv.1912.13477,
title = {Interaction laws of monads and comonads},
author = {Shin-ya Katsumata and Exequiel Rivas and Tarmo Uustalu},
journal= {arXiv preprint arXiv:1912.13477},
year = {2020}
}