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-Inferred
Language	Haskell2010

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 generated 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

data Density k a where
- Density :: (k b -> a) -> k b -> Density k a
liftDensity :: Comonad w => w a -> Density w a
densityToAdjunction :: Adjunction f g => Density f a -> f (g a)
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

Documentation

data Density k a where #

Constructors

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

Instances

Instances details

ComonadTrans (Density :: (Type -> Type) -> Type -> Type) #
Instance details Defined in Control.Comonad.Density Methods lower :: Comonad w => Density w a -> w a #
Functor (Density f) #
Instance details Defined in Control.Comonad.Density Methods fmap :: (a -> b) -> Density f a -> Density f b # (<$) :: a -> Density f b -> Density f a #
Applicative f => Applicative (Density f) #
Instance details Defined in Control.Comonad.Density 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 f) #
Instance details Defined in Control.Comonad.Density 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) #
Instance details Defined in Control.Comonad.Density 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 f) #
Instance details Defined in Control.Comonad.Density 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