如何证明monad是一个仿函数和一个应用函子？

Question

Monads are known to be theoretically a subset of functors and specifically applicative functors, even though it's not indicated in Haskell's type system.

Knowing that, given a monad and basing on return and bind , how to:

• derive fmap ,
• derive <*> ?

Answer 1

Well, fmap is just (a -> b) -> fa -> fb , ie we want to transform the monadic action's result with a pure function. That's easy to write with do notation:

fmap f m = do
  a <- m
  return (f a)

or, written "raw":

fmap f m = m >>= \a -> return (f a)

This is available as Control.Monad.liftM .

pure :: a -> fa is of course return . (<*>) :: f (a -> b) -> fa -> fb is a little trickier. We have an action returning a function, and an action returning its argument, and we want an action returning its result. In do notation again:

mf <*> mx = do
  f <- mf
  x <- mx
  return (f x)

Or, desugared:

mf <*> mx =
  mf >>= \f ->
  mx >>= \x ->
  return (f x)

Tada! This is available as Control.Monad.ap , so we can give a complete instance of Functor and Applicative for any monad M as follows:

instance Functor M where
  fmap = liftM

instance Applicative M where
  pure = return
  (<*>) = ap

Ideally, we'd be able to specify these implementations directly in Monad , to relieve the burden of defining separate instances for every monad, such as with this proposal . If that happens, there'll be no real obstacle to making Applicative a superclass of Monad , as it'll ensure it doesn't break any existing code. On the other hand, this means that the boilerplate involved in defining Functor and Applicative instances for a given Monad is minimal, so it's easy to be a "good citizen" (and such instances should be defined for any monad).

Answer 2

fmap = liftM and (<*>) = ap . Here are links to the source code for liftM and ap . I presume you know how to desugar do notation.