Breaking a monad-comonad symmetry between computational effects

Jean-Guillaume Dumas (1) , Dominique Duval (1) , Jean-Claude Reynaud (2)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 RC - Reynaud Consulting

Computational effects may often be interpreted in the Kleisli category of a monad or in the coKleisli category of a comonad. The duality between monads and comonads corresponds, in general, to a symmetry between construction and observation, for instance between raising an exception and looking up a state. Thanks to the properties of adjunction one may go one step further: the coKleisli-on-Kleisli category of a monad provides a kind of observation with respect to a given construction, while dually the Kleisli-on-coKleisli category of a comonad provides a kind of construction with respect to a given observation. In the previous examples this gives rise to catching an exception and updating a state. However, the interpretation of computational effects is usually based on a category which is not self-dual, like the category of sets. This leads to a breaking of the monad-comonad duality. For instance, in a distributive category the state effect has much better properties than the exception effect. This remark provides a novel point of view on the usual mechanism for handling exceptions. The aim of this paper is to build an equational semantics for handling exceptions based on the coKleisli-on-Kleisli category of the monad of exceptions. We focus on n-ary functions and conditionals. We propose a programmer's language for exceptions and we prove that it has the required behaviour with respect to n-ary functions and conditionals.

Mots clés: Computational effects; monads and comonads; duality; decorated logics

Domaines: Logique en informatique [cs.LO]; Catégories et ensembles [math.CT]

Jean-Guillaume Dumas, Dominique Duval, Jean-Claude Reynaud. Breaking a monad-comonad symmetry between computational effects. [Research Report] arXiV. 2014. ⟨hal-00941653⟩