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.Monoidal
import qualified Data.Category.Adjunction as A
data Kleisli ((~>) :: * -> * -> *) m a b where
Kleisli :: Monad m -> Obj (~>) b -> a ~> (m :% b) -> Kleisli (~>) m a b
kleisliId :: (Category (~>), Functor m, Dom m ~ (~>), Cod m ~ (~>))
=> Monad m -> Obj (~>) a -> Kleisli (~>) m a a
kleisliId m a = Kleisli m a $ unit m ! a
instance (Category (~>), Functor m, Dom m ~ (~>), Cod m ~ (~>)) => Category (Kleisli (~>) m) where
src (Kleisli m _ f) = kleisliId m (src f)
tgt (Kleisli m b _) = kleisliId m b
(Kleisli m c f) . (Kleisli _ _ g) = Kleisli m c $ (multiply m ! c) . (monadFunctor m % f) . g
data KleisliAdjF ((~>) :: * -> * -> *) m where
KleisliAdjF :: Monad m -> KleisliAdjF (~>) m
type instance Dom (KleisliAdjF (~>) m) = (~>)
type instance Cod (KleisliAdjF (~>) m) = Kleisli (~>) m
type instance KleisliAdjF (~>) m :% a = a
instance (Category (~>), Functor m, Dom m ~ (~>), Cod m ~ (~>)) => Functor (KleisliAdjF (~>) m) where
KleisliAdjF m % f = Kleisli m (tgt f) $ (unit m ! tgt f) . f
data KleisliAdjG ((~>) :: * -> * -> *) m where
KleisliAdjG :: Monad m -> KleisliAdjG (~>) m
type instance Dom (KleisliAdjG (~>) m) = Kleisli (~>) m
type instance Cod (KleisliAdjG (~>) m) = (~>)
type instance KleisliAdjG (~>) m :% a = m :% a
instance (Category (~>), Functor m, Dom m ~ (~>), Cod m ~ (~>)) => Functor (KleisliAdjG (~>) m) where
KleisliAdjG m % Kleisli _ b f = (multiply m ! b) . (monadFunctor m % f)
kleisliAdj :: (Functor m, Dom m ~ (~>), Cod m ~ (~>), Category (~>))
=> Monad m -> A.Adjunction (Kleisli (~>) m) (~>) (KleisliAdjF (~>) m) (KleisliAdjG (~>) m)
kleisliAdj m = A.mkAdjunction (KleisliAdjF m) (KleisliAdjG m)
(\x -> unit m ! x)
(\(Kleisli _ x _) -> Kleisli m x $ monadFunctor m % x)