module Data.Category.Kleisli where
import Prelude hiding ((.), id, Functor(..), Monad(..))
import Data.Category
import Data.Category.Functor
import Data.Category.NaturalTransformation
import Data.Category.Adjunction
class Functor m => Pointed m where
point :: m -> Id (Dom m) :~> m
class Pointed m => Monad m where
join :: m -> (m :.: m) :~> m
data Kleisli ((~>) :: * -> * -> *) m a b where
Kleisli :: m -> Obj (~>) b -> a ~> (m :% b) -> Kleisli (~>) m a b
instance (Category (~>), Monad m, Dom m ~ (~>), Cod m ~ (~>)) => Category (Kleisli (~>) m) where
data Obj (Kleisli (~>) m) a = KleisliO m (Obj (~>) a)
src (Kleisli m _ f) = KleisliO m (src f)
tgt (Kleisli m b _) = KleisliO m b
id (KleisliO m o) = Kleisli m o $ point m ! o
(Kleisli m c f) . (Kleisli _ _ g) = Kleisli m c $ (join m ! c) . (m % f) . g
data KleisliAdjF ((~>) :: * -> * -> *) m where
KleisliAdjF :: (Category (~>), Monad m, Dom m ~ (~>), Cod m ~ (~>)) => m -> KleisliAdjF (~>) m
type instance Dom (KleisliAdjF (~>) m) = (~>)
type instance Cod (KleisliAdjF (~>) m) = Kleisli (~>) m
type instance KleisliAdjF (~>) m :% a = a
instance Functor (KleisliAdjF (~>) m) where
KleisliAdjF m %% x = KleisliO m x
KleisliAdjF m % f = Kleisli m (tgt f) $ (point m ! tgt f) . f
data KleisliAdjG ((~>) :: * -> * -> *) m where
KleisliAdjG :: (Category (~>), Monad m, Dom m ~ (~>), Cod m ~ (~>)) => m -> KleisliAdjG (~>) m
type instance Dom (KleisliAdjG (~>) m) = Kleisli (~>) m
type instance Cod (KleisliAdjG (~>) m) = (~>)
type instance KleisliAdjG (~>) m :% a = m :% a
instance Functor (KleisliAdjG (~>) m) where
KleisliAdjG m %% KleisliO _ x = m %% x
KleisliAdjG m % Kleisli _ b f = (join m ! b) . (m % f)
kleisliAdj :: (Monad m, Dom m ~ (~>), Cod m ~ (~>), Category (~>))
=> m -> Adjunction (Kleisli (~>) m) (~>) (KleisliAdjF (~>) m) (KleisliAdjG (~>) m)
kleisliAdj m = mkAdjunction (KleisliAdjF m) (KleisliAdjG m)
(\x -> point m ! x)
(\(KleisliO _ x) -> Kleisli m x $ m % id x)