Categorical.hs

{-# LANGUAGE GADTs, FlexibleInstances, MultiParamTypeClasses #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Data.Monoid.Categorical
-- Copyright   :  (c) Edward Kmett 2009
-- License     :  BSD-style
-- Maintainer  :  libraries@haskell.org
-- Stability   :  experimental
-- Portability :  portable
--
-----------------------------------------------------------------------------

module Data.Monoid.Categorical
    ( module Data.Monoid.Reducer
    , module Control.Category
    -- * Generalized Endo
    , GEndo(GEndo, getGEndo)
    -- * Monoids as Categories
    , Mon(Mon)
    , getMon
    ) where

import Prelude hiding ((.),id)
import Data.Monoid.Reducer
import Control.Category

-- | The 'Monoid' of the endomorphisms over some object in an arbitrary 'Category'.
data GEndo k a = GEndo { getGEndo :: k a a } 

instance Category k =>  Monoid (GEndo k a) where
    mempty = GEndo id
    GEndo f `mappend` GEndo g = GEndo (f . g)

-- | A 'Monoid' is just a 'Category' with one object. 
data Mon m n o where
    Mon :: Monoid m => m -> Mon m a a

-- | Extract the 'Monoid' from its representation as a 'Category'
getMon :: Mon m m m -> m 
getMon (Mon m) = m

instance Monoid m => Category (Mon m) where
    id = Mon mempty
    Mon a . Mon b = Mon (a `mappend` b)

instance Monoid m => Monoid (Mon m m m) where
    mempty = id
    mappend = (.)

instance (c `Reducer` m) => Reducer c (Mon m m m) where
    unit = Mon . unit