module Data.Category.Adjunction where
import Prelude hiding ((.), id, Functor)
import Control.Monad.Instances()
import Data.Category
import Data.Category.Functor
import Data.Category.NaturalTransformation
import Data.Category.Limit
data Adjunction c d f g where
Adjunction :: (Functor f, Functor g, Category c, Category d, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) =>
f -> g -> Nat d d (Id d) (g :.: f) -> Nat c c (f :.: g) (Id c) -> Adjunction c d f g
mkAdjunction :: (Functor f, Functor g, Category c, Category d, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d)
=> f -> g
-> (forall a. Obj d a -> Component (Id d) (g :.: f) a)
-> (forall a. Obj c a -> Component (f :.: g) (Id c) a)
-> Adjunction c d f g
mkAdjunction f g un coun = Adjunction f g (Nat Id (g :.: f) un) (Nat (f :.: g) Id coun)
unit :: Adjunction c d f g -> Id d :~> (g :.: f)
unit (Adjunction _ _ u _) = u
counit :: Adjunction c d f g -> (f :.: g) :~> Id c
counit (Adjunction _ _ _ c) = c
leftAdjunct :: Adjunction c d f g -> Obj d a -> c (f :% a) b -> d a (g :% b)
leftAdjunct (Adjunction _ g un _) i h = (g % h) . (un ! i)
rightAdjunct :: Adjunction c d f g -> Obj c b -> d a (g :% b) -> c (f :% a) b
rightAdjunct (Adjunction f _ _ coun) i h = (coun ! i) . (f % h)
adjunctionInitialProp :: Adjunction c d f g -> Obj d y -> InitialUniversal y g (f :% y)
adjunctionInitialProp adj@(Adjunction f _ un _) y = InitialUniversal (f %% y) (un ! y) (rightAdjunct adj)
adjunctionTerminalProp :: Adjunction c d f g -> Obj c x -> TerminalUniversal x f (g :% x)
adjunctionTerminalProp adj@(Adjunction _ g _ coun) x = TerminalUniversal (g %% x) (coun ! x) (leftAdjunct adj)
initialPropAdjunction :: (Functor f, Functor g, Category c, Category d, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d)
=> f -> g -> (forall y. Obj d y -> InitialUniversal y g (f :% y)) -> Adjunction c d f g
initialPropAdjunction f g univ = mkAdjunction f g un coun
where
coun a = let ga = g %% a in initialFactorizer (univ ga) a (id ga)
un a = initialMorphism (univ a)
terminalPropAdjunction :: (Functor f, Functor g, Category c, Category d, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d)
=> f -> g -> (forall x. Obj c x -> TerminalUniversal x f (g :% x)) -> Adjunction c d f g
terminalPropAdjunction f g univ = mkAdjunction f g un coun
where
un a = let fa = f %% a in terminalFactorizer (univ fa) a (id fa)
coun a = terminalMorphism (univ a)
data AdjArrow c d where
AdjArrow :: (Category c, Category d) => Adjunction c d f g -> AdjArrow (CatW c) (CatW d)
instance Category AdjArrow where
data Obj AdjArrow a where
AdjCategory :: Category (~>) => Obj AdjArrow (CatW (~>))
src (AdjArrow _) = AdjCategory
tgt (AdjArrow _) = AdjCategory
id AdjCategory = AdjArrow $ mkAdjunction Id Id id id
AdjArrow (Adjunction f g u c) . AdjArrow (Adjunction f' g' u' c') = AdjArrow $
Adjunction (f' :.: f) (g :.: g') (wrap g f u' . u) (c' . cowrap f' g' c)
wrap :: (Functor g, Functor f, Dom g ~ Dom f', Dom g ~ Cod f)
=> g -> f -> Nat (Dom f') (Dom f') (Id (Dom f')) (g' :.: f') -> (g :.: f) :~> ((g :.: g') :.: (f' :.: f))
wrap g f (Nat Id (g' :.: f') n) = Nat (g :.: f) ((g :.: g') :.: (f' :.: f)) $ (g %) . n . (f %%)
cowrap :: (Functor f', Functor g', Dom f' ~ Dom g, Dom f' ~ Cod g')
=> f' -> g' -> Nat (Dom g) (Dom g) (f :.: g) (Id (Dom g)) -> ((f' :.: f) :.: (g :.: g')) :~> (f' :.: g')
cowrap f' g' (Nat (f :.: g) Id n) = Nat ((f' :.: f) :.: (g :.: g')) (f' :.: g') $ (f' %) . n . (g' %%)
curryAdj :: Adjunction (->) (->) (EndoHask ((,) e)) (EndoHask ((->) e))
curryAdj = mkAdjunction EndoHask EndoHask (\HaskO -> \a e -> (e, a)) (\HaskO -> \(e, f) -> f e)
limitAdj :: forall j (~>). HasLimits j (~>)
=> LimitFunctor j (~>)
-> Adjunction (Nat j (~>)) (~>) (Diag j (~>)) (LimitFunctor j (~>))
limitAdj LimitFunctor = terminalPropAdjunction Diag LimitFunctor univ
where
univ :: Obj (Nat j (~>)) f -> TerminalUniversal f (Diag j (~>)) (LimitFam j (~>) f)
univ f @ NatO{} = limitUniv f
colimitAdj :: forall j (~>). HasColimits j (~>)
=> ColimitFunctor j (~>)
-> Adjunction (~>) (Nat j (~>)) (ColimitFunctor j (~>)) (Diag j (~>))
colimitAdj ColimitFunctor = initialPropAdjunction ColimitFunctor Diag univ
where
univ :: Obj (Nat j (~>)) f -> InitialUniversal f (Diag j (~>)) (ColimitFam j (~>) f)
univ f @ NatO{} = colimitUniv f