The category of monads

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BSD3 licensed by Dan Burton

MHask is the category where

  • The objects are Haskell types of kind (* → *) that have an instance of Prelude.Monad

  • An arrow from object m to object n is a Haskell function of the form (forall x. m x → n x)

  • Arrow composition is merely a specialization of Haskell function composition

  • The identity arrow for the object m is the function in Haskell, specialized to (forall x. m x → m x)

It is assumed that any instances of the classes provided here also have an instance for Monad of the form instance (Monad m) ⇒ Monad (t m), thus guaranteeing that (t m) (or (t i j m) for the indexed types) is always a Monad. If it were valid Haskell, I'd write:

class (forall m. Monad m ⇒ Monad (t m)) ⇒ Functor t

MHask.Functor.Functor should actually be called Endofunctor, because if m is an object in MHask, then an instance t of Functor can map object m to object (t m), and arrows from m to n get mapped to arrows from (t m) to (t n).

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