{-# LANGUAGE MultiParamTypeClasses #-} ------------------------------------------------------------------------------------------- -- | -- Module : Control.Category.Associative -- Copyright : 2008 Edward Kmett -- License : BSD -- -- Maintainer : Edward Kmett <ekmett@gmail.com> -- 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(..) ) where import Control.Categorical.Bifunctor {- | A category with an associative bifunctor satisfying Mac Lane\'s pentagonal coherence identity law: > bimap id associate . associate . bimap associate id = associate . associate > bimap disassociate id . disassociate . bimap id disassociate = disassociate . disassociate -} class Bifunctor p k k k => Associative k p where associate :: k (p (p a b) c) (p a (p b c)) disassociate :: k (p a (p b c)) (p (p a b) c) {-- RULES "copentagonal coherence" first disassociate . disassociate . second disassociate = disassociate . disassociate "pentagonal coherence" second associate . associate . first associate = associate . associate --} instance Associative (->) (,) where associate ((a,b),c) = (a,(b,c)) disassociate (a,(b,c)) = ((a,b),c) instance Associative (->) Either where associate (Left (Left a)) = Left a associate (Left (Right b)) = Right (Left b) associate (Right c) = Right (Right c) disassociate (Left a) = Left (Left a) disassociate (Right (Left b)) = Left (Right b) disassociate (Right (Right c)) = Right c