Monad package:classy-prelude

The Monad class defines the basic operations over a monad, a concept from a branch of mathematics known as category theory. From the perspective of a Haskell programmer, however, it is best to think of a monad as an abstract datatype of actions. Haskell's do expressions provide a convenient syntax for writing monadic expressions. Instances of Monad should satisfy the following:

• Left identity return a >>= k = k a
• Right identity m >>= return = m
• Associativity m >>= (\x -> k x >>= h) = (m >>= k) >>= h

Furthermore, the Monad and Applicative operations should relate as follows:

• pure = return

• m1 <*> m2 = m1 >>= (x1 -> m2
  >>= (x2 -> return (x1 x2)))

The above laws imply:

• fmap f xs = xs >>= return .
  f

• (>>) = (*>)

and that pure and (<*>) satisfy the applicative functor laws. The instances of Monad for lists, Maybe and IO defined in the Prelude satisfy these laws.

Monads in which IO computations may be embedded. Any monad built by applying a sequence of monad transformers to the IO monad will be an instance of this class. Instances should satisfy the following laws, which state that liftIO is a transformer of monads:

• liftIO . return = return

• liftIO (m >>= f) = liftIO m >>=
  (liftIO . f)

Monads that also support choice and failure.

See examples in Control.Monad.Reader. Note, the partially applied function type (->) r is a simple reader monad. See the instance declaration below.