Haskell - important typeclasses and cotypeclasses

typeclass.hs

-- none of this is terribly interesting, mostly here for reference 

class Functor (f :: * -> *) where
  fmap :: (a -> b) -> f a -> f b
  (<$) :: a -> f b -> f a

-- The same as Functor! 
-- http://stackoverflow.com/questions/34732571/why-there-is-no-cofunctor-typeclass-in-haskell/34732721
class Cofunctor (f :: * -> *) where
  fmap :: (b -> a) -> f b -> f a
  (<$) :: b -> f a -> f b
  
class Functor f => Applicative (f :: * -> *) where
  pure  :: a -> f a
  (<*>) :: f (a -> b) -> f a -> f b
  (*>)  :: f a -> f b -> f b
  (<*)  :: f a -> f b -> f a

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

class Functor w => Comonad w where
  extract :: w a -> a
  duplicate :: w a -> w (w a)
  extend :: (w a -> b) -> w a -> w b
  
class Monoid a where
  mempty :: a
  mappend :: a -> a -> a

-- http://stackoverflow.com/questions/23855070/what-does-a-nontrivial-comonoid-look-like  
class Comonoid m where
  destroy :: m -> ()
  split   :: m -> (m, m)