category-extras-0.53.6: Various modules and constructs inspired by category theory Contents Index

Control.Comonad.Density Portability non-portable (rank-2 polymorphism) Stability experimental Maintainer Edward Kmett <ekmett@gmail.com>

Description

The density comonad for a functor. aka the comonad cogenerated by a functor The ''density'' term dates back to Dubuc''s 1974 thesis. The term ''monad genererated by a functor'' dates back to 1972 in Street''s ''Formal Theory of Monads''.

Synopsis

data Density k a = forall b . Density (k b -> a) (k b) densityToLan :: Density k a -> Lan k k a lanToDensity :: Lan k k a -> Density k a toDensity :: Functor s => (forall a. k a -> s (k a)) -> Density k :~> s fromDensity :: (Density k :~> s) -> k a -> s (k a) liftDensity :: Comonad w => w a -> Density w a lowerDensity :: Comonad w => Density w a -> w a densityToAdjunction :: Adjunction f g => Density f a -> f (g a) adjunctionToDensity :: Adjunction f g => f (g a) -> Density f a densityToComposedAdjunction :: (Composition o, Adjunction f g) => Density f :~> (f `o` g) composedAdjunctionToDensity :: (Composition o, Adjunction f g) => (f `o` g) :~> Density f improveCofree :: Functor f => (forall w. ComonadCofree f w => w a) -> Cofree f a

Documentation

data Density k a

Constructors forall b . Density (k b -> a) (k b) Instances ComonadTrans Density ComonadContext e w => ComonadContext e (Density w) ComonadReader e w => ComonadReader e (Density w) RunComonadCofree f w => RunComonadCofree f (Density w) ComonadCofree f w => ComonadCofree f (Density w) Functor (Density f) Copointed (Density f) Comonad (Density f)

densityToLan :: Density k a -> Lan k k a

lanToDensity :: Lan k k a -> Density k a

toDensity :: Functor s => (forall a. k a -> s (k a)) -> Density k :~> s

Nat(k, s.k) is isomorphic to Nat (Density k, s) (forwards)

fromDensity :: (Density k :~> s) -> k a -> s (k a)

Nat(k, s.k) is isomorphic to Nat (Density k, s) (backwards)

liftDensity :: Comonad w => w a -> Density w a

The natural isomorphism between a comonad w and the comonad generated by w (forwards).

lowerDensity :: Comonad w => Density w a -> w a

The natural isomorphism between a comonad w and the comonad generated by w (backwards).

densityToAdjunction :: Adjunction f g => Density f a -> f (g a)

adjunctionToDensity :: Adjunction f g => f (g a) -> Density f a

densityToComposedAdjunction :: (Composition o, Adjunction f g) => Density f :~> (f `o` g)

composedAdjunctionToDensity :: (Composition o, Adjunction f g) => (f `o` g) :~> Density f

improveCofree :: Functor f => (forall w. ComonadCofree f w => w a) -> Cofree f a

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