2019/12/31 by Katsumata, Shin-ya, Rivas, Exequiel, Uustalu, Tarmo
#18C50 #68Q55 #Category Theory (math.CT) #F.3.2 #FOS: Computer and information sciences #FOS: Mathematics #Logic in Computer Science (cs.LO) #Programming Languages (cs.PL)
paper · doi:10.48550/arxiv.1912.13477
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.