如何将一个 comonad 和一个 monad 组合成一个 comonad？

Question

Assume I have

• a comonad D
• a monad T
• a distributive law l: DT -> TD of the comonad D over the monad T .

How can I define the comonad DT ?

Answer 1

You can't. Suppose D is the identity comonad and T is Cont Void , ie the continuation monad at the empty type.

newtype D a = D {runD :: a}
newtype T a = T {runT :: (a -> Void) -> Void}

Then distributivity holds trivially. But extract:: D (T a) -> a is not definable as a total computable program. It would be double negation elimination forall a. ((a -> Void) -> Void) -> a forall a. ((a -> Void) -> Void) -> a , which is not definable in constructive languages.