Adjunction.hs

{-# LANGUAGE Rank2Types, MultiParamTypeClasses, FunctionalDependencies, UndecidableInstances #-}

-------------------------------------------------------------------------------------------
-- |
-- Module	: Data.Functor.Adjunction
-- Copyright 	: 2008-2011 Edward Kmett
-- License	: BSD
--
-- Maintainer	: Edward Kmett <ekmett@gmail.com>
-- Stability	: experimental
-- Portability	: rank 2 types, MPTCs, fundeps
--
-------------------------------------------------------------------------------------------
module Data.Functor.Adjunction 
  ( Adjunction(..)
  , distributeAdjunct
  , Representation(..)
  , repAdjunction
  ) where

import Control.Applicative
import Control.Monad.Instances ()
import Control.Monad.Trans.Identity

import Control.Monad.Trans.Reader
import Control.Comonad.Trans.Env

import Data.Distributive
import Data.Functor.Identity
import Data.Functor.Compose
-- import qualified Data.Functor.Contravariant.Adjunction as C
-- import qualified Data.Functor.Contravariant.Compose as C

-- | An adjunction between Hask and Hask.
--
-- > rightAdjunct unit = id
-- > leftAdjunct counit = id 
class (Functor f, Distributive g) => Adjunction f g | f -> g, g -> f where
  unit :: a -> g (f a)
  counit :: f (g a) -> a
  leftAdjunct :: (f a -> b) -> a -> g b
  rightAdjunct :: (a -> g b) -> f a -> b
  
  unit = leftAdjunct id
  counit = rightAdjunct id
  leftAdjunct f = fmap f . unit
  rightAdjunct f = counit . fmap f

-- | Every right adjoint is representable by its left adjoint applied to unit 
-- Consequently, we use the isomorphism from ((->) f ()) ~ g to distribute
-- the right adjoint over any other functor.
distributeAdjunct :: (Adjunction f g, Functor w) => w (g a) -> g (w a)
distributeAdjunct wg = leftAdjunct (\a -> fmap (\b -> rightAdjunct (const b) a) wg) ()

instance Adjunction ((,)e) ((->)e) where
  leftAdjunct f a e = f (e, a)
  rightAdjunct f ~(e, a) = f a e

instance Adjunction Identity Identity where
  leftAdjunct f = Identity . f . Identity
  rightAdjunct f = runIdentity . f . runIdentity

instance Adjunction f g => Adjunction (IdentityT f) (IdentityT g) where
  unit = IdentityT . leftAdjunct IdentityT
  counit = rightAdjunct runIdentityT . runIdentityT

instance Adjunction w m => Adjunction (EnvT e w) (ReaderT e m) where
  unit a = ReaderT $ \e -> EnvT e <$> unit a
  counit (EnvT e w) = counit $ fmap (flip runReaderT e) w

instance (Adjunction f g, Adjunction f' g') => Adjunction (Compose f' f) (Compose g g') where
  unit = Compose . leftAdjunct (leftAdjunct Compose) 
  counit = rightAdjunct (rightAdjunct getCompose) . getCompose

data Representation f x = Representation
  { rep :: forall a. (x -> a) -> f a
  , unrep :: forall a. f a -> x -> a
  }
 
repAdjunction :: Adjunction f g => Representation g (f ())
repAdjunction = Representation 
  { rep = flip leftAdjunct ()
  , unrep = rightAdjunct . const
  }