{-# LANGUAGE TypeFamilies, MultiParamTypeClasses #-} ------------------------------------------------------------------------------------------- -- | -- Module : Control.Category.Monoidal -- Copyright : 2008 Edward Kmett -- License : BSD -- -- Maintainer : Edward Kmett <ekmett@gmail.com> -- Stability : experimental -- Portability : non-portable (class-associated types) -- -- A 'Monoidal' category is a category with an associated biendofunctor that has an identity, -- which satisfies Mac Lane''s pentagonal and triangular coherence conditions -- Technically we usually say that category is 'Monoidal', but since -- most interesting categories in our world have multiple candidate bifunctors that you can -- use to enrich their structure, we choose here to think of the bifunctor as being -- monoidal. This lets us reuse the same 'Bifunctor' over different categories without -- painful newtype wrapping. -- The use of class associated types here makes Control.Category.Cartesian FAR more palatable ------------------------------------------------------------------------------------------- module Control.Category.Monoidal ( Id , Monoidal(..) , Comonoidal(..) ) where import Control.Category.Associative import Data.Void -- | Denotes that we have some reasonable notion of 'Identity' for a particular 'Bifunctor' in this 'Category'. This -- notion is currently used by both 'Monoidal' and 'Comonoidal' type family Id k p :: * {- | A monoidal category. 'idl' and 'idr' are traditionally denoted lambda and rho the triangle identity holds: > first idr = second idl . associate > second idl = first idr . associate -} class Associative k p => Monoidal k p where idl :: k (p (Id k p) a) a idr :: k (p a (Id k p)) a {- | A comonoidal category satisfies the dual form of the triangle identities > first idr = disassociate . second idl > second idl = disassociate . first idr This type class is also (ab)used for the inverse operations needed for a strict (co)monoidal category. A strict (co)monoidal category is one that is both 'Monoidal' and 'Comonoidal' and satisfies the following laws: > idr . coidr = id > idl . coidl = id > coidl . idl = id > coidr . idr = id -} class Disassociative k p => Comonoidal k p where coidl :: k a (p (Id k p) a) coidr :: k a (p a (Id k p)) {-- RULES -- "bimap id idl/associate" second idl . associate = first idr -- "bimap idr id/associate" first idr . associate = second idl -- "disassociate/bimap id idl" disassociate . second idl = first idr -- "disassociate/bimap idr id" disassociate . first idr = second idl "idr/coidr" idr . coidr = id "idl/coidl" idl . coidl = id "coidl/idl" coidl . idl = id "coidr/idr" coidr . idr = id "idr/braid" idr . braid = idl "idl/braid" idl . braid = idr "braid/coidr" braid . coidr = coidl "braid/coidl" braid . coidl = coidr --} type instance Id (->) (,) = () type instance Id (->) Either = Void instance Monoidal (->) (,) where idl = snd idr = fst instance Monoidal (->) Either where idl = either absurd id idr = either id absurd instance Comonoidal (->) (,) where coidl a = ((),a) coidr a = (a,()) instance Comonoidal (->) Either where coidl = Right coidr = Left