Data.Category.Adjunction

Adjunctions

data Adjunction c d f g

Constructors

(Functor f, Functor g, Category c, Category d, Dom f ~ d, Cod f ~ c, Dom g ~ c, Cod g ~ d) => Adjunction

Fields

leftAdjoint :: f

rightAdjoint :: g

unit :: Nat d d (Id d) (g :.: f)

counit :: Nat c c (f :.: g) (Id c)

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

leftAdjunct :: Adjunction c d f g -> Obj d a -> c (f :% a) b -> d a (g :% b)

rightAdjunct :: Adjunction c d f g -> Obj c b -> d a (g :% b) -> c (f :% a) b

Adjunctions as a category

idAdj :: Category k => Adjunction k k (Id k) (Id k)

composeAdj :: Adjunction d e f g -> Adjunction c d f' g' -> Adjunction c e (f' :.: f) (g :.: g')

data AdjArrow c d where

Constructors

AdjArrow :: (Category c, Category d) => Adjunction c d f g -> AdjArrow (CatW c) (CatW d)

Instances

Category AdjArrow

The category with categories as objects and adjunctions as arrows.

Adjunctions from universal morphisms

initialPropAdjunction :: forall f g c d. (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

terminalPropAdjunction :: forall f g c d. (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

Universal morphisms from adjunctions

adjunctionInitialProp :: Adjunction c d f g -> Obj d y -> InitialUniversal y g (f :% y)

adjunctionTerminalProp :: Adjunction c d f g -> Obj c x -> TerminalUniversal x f (g :% x)

Examples

precomposeAdj :: Category e => Adjunction c d f g -> Adjunction (Nat c e) (Nat d e) (Precompose g e) (Precompose f e)

postcomposeAdj :: Category e => Adjunction c d f g -> Adjunction (Nat e c) (Nat e d) (Postcompose f e) (Postcompose g e)

contAdj :: Adjunction (Op (->)) (->) (Opposite ((->) :-*: r) :.: OpOpInv (->)) ((->) :-*: r)