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

Methods

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

Functor (Density f)

Methods

fmap :: (a -> b) -> Density f a -> Density f b

(<$) :: a -> Density f b -> Density f a

Applicative f => Applicative (Density f)

Methods

pure :: a -> Density f a

(<*>) :: Density f (a -> b) -> Density f a -> Density f b

(*>) :: Density f a -> Density f b -> Density f b

(<*) :: Density f a -> Density f b -> Density f a

Comonad (Density f)

Methods

extract :: Density f a -> a

duplicate :: Density f a -> Density f (Density f a)

extend :: (Density f a -> b) -> Density f a -> Density f b

Apply f => Apply (Density f)

Methods

(<.>) :: Density f (a -> b) -> Density f a -> Density f b

(.>) :: Density f a -> Density f b -> Density f b

(<.) :: Density f a -> Density f b -> Density f a

Extend (Density f)

Methods

duplicated :: Density f a -> Density f (Density f a)

extended :: (Density f a -> b) -> Density f a -> Density f b

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