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Breaking a monad-comonad symmetry between computational effects
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.Comment: arXiv admin note: substantial text overlap with arXiv:1310.060