haskell - Are there contravariant monads?

Question

Functors can be covariant and contravariant. Can this covariant/contravariant duality also be applied to monads?

Something like:

class Monad m where
  return :: a -> m a
  (>>=) :: m a -> (a -> m b) -> m b    

class ContraMonad m where
  return :: a -> m a
  contrabind :: m a -> (b -> m a) -> m b

Does ContraMonad class make sense? Any examples?

Answer 1

Well, of course, it's possible to define it, but I doubt it would be of any use.

There is a popular saying that "monad is just a monoid in a category of endofunctors". What it means is, first of all, that we have a category of endofunctors (meaning, (covariant) functors from some category to itself), and what's more, we have some multiplication on this endofunctors (in this case — composition). And then monad fits into some general framework that we don't have to worry about right now. The point is, there is no "multiplication" of contravariant functors. Composition of two covariant functors is again a covariant functor; but composition of two contravariant functors is not a contravariant functor (rather it's a covariant functor, so, a totally different beast).

So, "contravariant monads" do not really make sense.