Data.Functor.Kan.Lan

Left Kan Extensions

Left Kan Extensions

data Lan g h a where

The left Kan extension of a Functor h along a Functor g.

Constructors

Lan :: (g b -> a) -> h b -> Lan g h a

Instances

Functor (Lan f g)

(Functor g, Applicative h) => Applicative (Lan g h)

(Functor g, Apply h) => Apply (Lan g h)

toLan :: Functor f => (forall a. h a -> f (g a)) -> Lan g h b -> f b

The universal property of a left Kan extension.

fromLan :: (forall a. Lan g h a -> f a) -> h b -> f (g b)

fromLan and toLan witness a (higher kinded) adjunction between Lan g and (Compose g)

toLan . fromLan ≡ id
fromLan . toLan ≡ id

glan :: h a -> Lan g h (g a)

This is the natural transformation that defines a Left Kan extension.

composeLan :: (Composition compose, Functor f) => Lan f (Lan g h) a -> Lan (compose f g) h a

composeLan and decomposeLan witness the natural isomorphism from Lan f (Lan g h) and Lan (f o g) h

composeLan . decomposeLan ≡ id
decomposeLan . composeLan ≡ id

decomposeLan :: Composition compose => Lan (compose f g) h a -> Lan f (Lan g h) a

adjointToLan :: Adjunction f g => g a -> Lan f Identity a

adjointToLan . lanToAdjoint ≡ id
lanToAdjoint . adjointToLan ≡ id

lanToAdjoint :: Adjunction f g => Lan f Identity a -> g a

composedAdjointToLan :: Adjunction f g => h (g a) -> Lan f h a

lanToComposedAdjoint :: (Functor h, Adjunction f g) => Lan f h a -> h (g a)

lanToComposedAdjoint and composedAdjointToLan witness the natural isomorphism between Lan f h and Compose h g given f -| g

composedAdjointToLan . lanToComposedAdjoint ≡ id
lanToComposedAdjoint . composedAdjointToLan ≡ id