Control/Category.hs

{-# LANGUAGE Trustworthy #-}
{-# LANGUAGE GADTs #-}
{-# LANGUAGE NoImplicitPrelude #-}
{-# LANGUAGE PolyKinds #-}

-----------------------------------------------------------------------------
-- |
-- Module : Control.Category
-- Copyright : (c) Ashley Yakeley 2007
-- License : BSD-style (see the LICENSE file in the distribution)
--
-- Maintainer : ashley@semantic.org
-- Stability : experimental
-- Portability : portable

-- http://ghc.haskell.org/trac/ghc/ticket/1773 

module Control.Category where

import qualified GHC.Base (id,(.))
import Data.Type.Coercion
import Data.Type.Equality
import GHC.Prim (coerce)

infixr 9 .
infixr 1 >>>, <<<

-- | A class for categories.
-- id and (.) must form a monoid.
class Category cat where
 -- | the identity morphism
 id :: cat a a

 -- | morphism composition
 (.) :: cat b c -> cat a b -> cat a c

{-# RULES
"identity/left" forall p .
 id . p = p
"identity/right" forall p .
 p . id = p
"association" forall p q r .
 (p . q) . r = p . (q . r)
 #-}

instance Category (->) where
 id = GHC.Base.id
 (.) = (GHC.Base..)

instance Category (:~:) where
 id = Refl
 Refl . Refl = Refl

instance Category Coercion where
 id = Coercion
 (.) Coercion = coerce

-- | Right-to-left composition
(<<<) :: Category cat => cat b c -> cat a b -> cat a c
(<<<) = (.)

-- | Left-to-right composition
(>>>) :: Category cat => cat a b -> cat b c -> cat a c
f >>> g = g . f