Copyright	(C) 2008-2016 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 k f) Source #
Methods fmap :: (a -> b) -> Density k f a -> Density k f b # (<$) :: a -> Density k f b -> Density k f a #
Applicative f => Applicative (Density * f) Source #
Methods pure :: a -> Density * f a # (<>) :: Density f (a -> b) -> Density * f a -> Density * f b # liftA2 :: (a -> b -> c) -> Density * f a -> Density * f b -> Density * f c # (>) :: Density f a -> Density * f b -> Density * f b # (<) :: Density f a -> Density * f b -> Density * f a #
Comonad (Density k f) Source #
Methods extract :: Density k f a -> a # duplicate :: Density k f a -> Density k f (Density k f a) # extend :: (Density k f a -> b) -> Density k f a -> Density k 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 # liftF2 :: (a -> b -> c) -> Density * f a -> Density * f b -> Density * f c #
Extend (Density k f) Source #
Methods duplicated :: Density k f a -> Density k f (Density k f a) # extended :: (Density k f a -> b) -> Density k f a -> Density k 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