Monoids.hs

-- monoids

type Unit = ()
type a × b = (a,b)
type a → b = (->)

class Monoid a where
  mempty :: Unit → a
  mappend :: a × a → a
  
-- Applicatives

data Day f g where
  Day :: (x -> y -> a) -> f x -> g y -> Day f g a
  
type Unit = Identity
type f × g = Day f g
type f → g = forall a. f a -> g a

class Applicative f where
  mempty :: Unit → f
  mappend :: f × f → f
  
-- Monoids

data Compose f g a = Compose (f (g a))
  
type Unit = Identity
type f × g = Compose f g
type f → g = forall a. f a -> g a

class Monad f where
  mempty :: Unit → f
  mappend :: f × f → f