-- {-# OPTIONS_GHC -fglasgow-exts -fallow-undecidable-instances #-}
-------------------------------------------------------------------------------------------
-- |
-- Module : Control.Category.Associative
-- Copyright : 2008 Edward Kmett
-- License : BSD
--
-- Maintainer : Edward Kmett
-- Stability : experimental
-- Portability : portable
--
-- NB: this contradicts another common meaning for an 'Associative' 'Category', which is one
-- where the pentagonal condition does not hold, but for which there is an identity.
--
-------------------------------------------------------------------------------------------
module Control.Category.Associative
( Associative(..)
, Coassociative(..)
) where
import Control.Functor
import Control.Category.Hask
{- | A category with an associative bifunctor satisfying Mac Lane\'s pentagonal coherence identity law:
> bimap id associate . associate . bimap associate id = associate . associate
-}
class Bifunctor p k k k => Associative k p where
associate :: k (p (p a b) c) (p a (p b c))
{- | A category with a coassociative bifunctor satisyfing the dual of Mac Lane's pentagonal coherence identity law:
> bimap coassociate id . coassociate . bimap id coassociate = coassociate . coassociate
-}
class Bifunctor s k k k => Coassociative k s where
coassociate :: k (s a (s b c)) (s (s a b) c)
{-# RULES
"copentagonal coherence" bimap coassociate id . coassociate . bimap id coassociate = coassociate . coassociate
"pentagonal coherence" bimap id associate . associate . bimap associate id = associate . associate
#-}
instance Associative Hask (,) where
associate ((a,b),c) = (a,(b,c))
instance Coassociative Hask (,) where
coassociate (a,(b,c)) = ((a,b),c)
instance Associative Hask Either where
associate (Left (Left a)) = Left a
associate (Left (Right b)) = Right (Left b)
associate (Right c) = Right (Right c)
instance Coassociative Hask Either where
coassociate (Left a) = Left (Left a)
coassociate (Right (Left b)) = Left (Right b)
coassociate (Right (Right c)) = Right c