Copyright	(C) 2008-2011 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable (GADTs, MPTCs)
Safe Haskell	Safe
Language	Haskell98

Control.Comonad.Density

Description

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.

Synopsis

Documentation

data Density k a where Source

Constructors

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

Instances

ComonadTrans Density Source
Methods lower :: Comonad w => Density w a -> w a
Functor (Density f) Source
Methods fmap :: (a -> b) -> Density f a -> Density f b (<$) :: a -> Density f b -> Density f a
Applicative f => Applicative (Density f) Source
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) Source
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) Source
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) Source
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 Source

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) Source

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 Source

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

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

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