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Interaction Laws of Monads and Comonads

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Abstract

We introduce and study functor-functor and monad-comonad interaction laws as math-ematical objects to describe interaction of effectful computations with behaviors of effect-performingmachines. 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 andSweedler dual, the greatest functor resp. monad interacting with a given functor or comonad is itsdual while the greatest comonad interacting with a given monad is its Sweedler dual. We relatemonad-comonad interaction laws to stateful runners. We show that functor-functor interaction lawsare Chu spaces over the category of endofunctors taken with the Day convolution monoidal struc-ture. Hasegawa’s glueing endows the category of these Chu spaces with a monoidal structure whosemonoid objects are monad-comonad interaction laws.

Dates and versions

hal-03112866 , version 1 (17-01-2021)

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Shin-Ya Katsumata, Exequiel Rivas, Tarmo Uustalu. Interaction Laws of Monads and Comonads. LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Jul 2020, Saarbrücken / Virtual, Germany. pp.604-618, ⟨10.1145/3373718.3394808⟩. ⟨hal-03112866⟩

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