Control.Comonad.Density

The Density Comonad for a Functor (aka the 'Comonad generated 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''.

The left Kan extension of a Functor along itself (Lan f f) forms a Comonad. This is that Comonad.

Documentation

data Density k a where

Constructors

Density :: (k b -> a) -> k b -> Density k a

Instances

ComonadTrans Density

Functor (Density f)

Applicative f => Applicative (Density f)

Comonad (Density f)

Apply f => Apply (Density f)

Extend (Density f)

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

The natural transformation from a Comonad w to the Comonad generated by w (forwards).

This is merely a right-inverse (section) of lower, rather than a full inverse.

 lower . liftDensity ≡ id

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

The Density Comonad of a left adjoint is isomorphic to the Comonad formed by that Adjunction.

This isomorphism is witnessed by densityToAdjunction and adjunctionToDensity.

 densityToAdjunction . adjunctionToDensity ≡ id
 adjunctionToDensity . densityToAdjunction ≡ id

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

densityToLan :: Density f a -> Lan f f a

lanToDensity :: Lan f f a -> Density f a

The Density Comonad of a Functor f is obtained by taking the left Kan extension (Lan) of f along itself. This isomorphism is witnessed by lanToDensity and densityToLan

 lanToDensity . densityToLan ≡ id
 densityToLan . lanToDensity ≡ id