{-# LANGUAGE TypeOperators, TypeFamilies, MultiParamTypeClasses, FlexibleInstances, FlexibleContexts, UndecidableInstances, RankNTypes #-}
-----------------------------------------------------------------------------
-- |
-- Module : Data.Category
-- Copyright : (c) Sjoerd Visscher 2010
-- License : BSD-style (see the file LICENSE)
--
-- Maintainer : sjoerd@w3future.com
-- Stability : experimental
-- Portability : non-portable
-----------------------------------------------------------------------------
module Data.Category (
-- * Categories
CategoryO(..)
, CategoryA(..)
, Apply(..)
-- * Functors
, F
, Dom
, Cod
, FunctorA(..)
, ContraFunctorA(..)
-- ** Functor instances
, Id(..)
, (:.:)(..)
, Const(..)
, (:*-:)(..)
, (:-*:)(..)
) where
import Prelude hiding ((.), id, ($))
-- | An instance CategoryO (~>) a declares a as an object of the category (~>).
class CategoryO (~>) a where
id :: a ~> a
-- | An instance CategoryA (~>) a b c defines composition of the arrows a ~> b and b ~> c.
class (CategoryO (~>) a, CategoryO (~>) b, CategoryO (~>) c) => CategoryA (~>) a b c where
(.) :: b ~> c -> a ~> b -> a ~> c
class (CategoryO (~>) a, CategoryO (~>) b) => Apply (~>) a b where
-- Would have liked to use ($) here, but that causes GHC to crash.
-- http://hackage.haskell.org/trac/ghc/ticket/3297
($$) :: a ~> b -> a -> b
-- | Functors are represented by a type tag. The type family 'F' turns the tag into the actual functor.
type family F ftag a :: *
-- | The domain, or source category, of the functor.
type family Dom ftag :: * -> * -> *
-- | The codomain, or target category, of the funcor.
type family Cod ftag :: * -> * -> *
-- | The mapping of arrows by covariant functors.
-- To make this type check, we need to pass the type tag along.
class (CategoryO (Dom ftag) a, CategoryO (Dom ftag) b)
=> FunctorA ftag a b where
(%) :: ftag -> Dom ftag a b -> Cod ftag (F ftag a) (F ftag b)
-- | The mapping of arrows by contravariant functors.
class (CategoryO (Dom ftag) a, CategoryO (Dom ftag) b)
=> ContraFunctorA ftag a b where
(-%) :: ftag -> Dom ftag a b -> Cod ftag (F ftag b) (F ftag a)
-- | The identity functor on (~>)
data Id ((~>) :: * -> * -> *) = Id
type instance F (Id (~>)) a = a
type instance Dom (Id (~>)) = (~>)
type instance Cod (Id (~>)) = (~>)
instance (CategoryO (~>) a, CategoryO (~>) b) => FunctorA (Id (~>)) a b where
Id % f = f
-- | The composition of two functors.
data (g :.: h) = g :.: h
type instance F (g :.: h) a = F g (F h a)
type instance Dom (g :.: h) = Dom h
type instance Cod (g :.: h) = Cod g
instance (FunctorA g (F h a) (F h b), FunctorA h a b, Cod h ~ Dom g) => FunctorA (g :.: h) a b where
(g :.: h) % f = g % (h % f)
-- | The constant functor.
data Const (c1 :: * -> * -> *) (c2 :: * -> * -> *) x = Const
type instance F (Const c1 c2 x) a = x
type instance Dom (Const c1 c2 x) = c1
type instance Cod (Const c1 c2 x) = c2
instance (CategoryO c1 a, CategoryO c1 b, CategoryO c2 x) => FunctorA (Const c1 c2 x) a b where
Const % f = id
-- | The covariant functor Hom(X,--)
data (x :*-: ((~>) :: * -> * -> *)) = HomX_
type instance F (x :*-: (~>)) a = x ~> a
type instance Dom (x :*-: (~>)) = (~>)
type instance Cod (x :*-: (~>)) = (->)
instance (CategoryO (~>) a, CategoryO (~>) b, CategoryA (~>) x a b) => FunctorA (x :*-: (~>)) a b where
HomX_ % f = (f .)
-- | The contravariant functor Hom(--,X)
data (((~>) :: * -> * -> *) :-*: x) = Hom_X
type instance F ((~>) :-*: x) a = a ~> x
type instance Dom ((~>) :-*: x) = (~>)
type instance Cod ((~>) :-*: x) = (->)
instance (CategoryO (~>) a, CategoryO (~>) b, CategoryA (~>) a b x) => ContraFunctorA ((~>) :-*: x) a b where
Hom_X -% f = (. f)