EndoCat.hs

{-# LANGUAGE DeriveGeneric #-}

import GHC.Generics
import Control.Category (Category)
import qualified Control.Category as Cat

-----------------------------------------------------------------------
--  Endofunctions
-----------------------------------------------------------------------

-- | The monoid of endofunctions under function composition.
-- This is the same as 'Endo'', specialized to the '(->)' category.
newtype Endo a = Endo { appEndo :: a -> a }
               deriving (Generic)

instance Monoid (Endo a) where
        mempty = Endo id
        Endo f `mappend` Endo g = Endo (f . g)

instance Semigroup (Endo a) where

-----------------------------------------------------------------------
--  Endomorphisms
-----------------------------------------------------------------------
        
-- | The monoid of endomorphisms under morphism composition.
newtype Endo' cat a = Endo' { appEndo' :: cat a a }
                    deriving Generic

instance Category cat => Monoid (Endo' cat a) where
        Endo' f `mappend` Endo' g = Endo' (f Cat.. g)
        mempty = Endo' Cat.id

instance Category cat => Semigroup (Endo' cat a) where