Copyright	(C) 2011-2013 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	provisional
Portability	MPTCs
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Functor.Contravariant.Adjunction

Description

Synopsis

class (Contravariant f, Representable g) => Adjunction (f :: Type -> Type) (g :: Type -> Type) | f -> g, g -> f where
- unit :: a -> g (f a)
- counit :: a -> f (g a)
- leftAdjunct :: (b -> f a) -> a -> g b
- rightAdjunct :: (a -> g b) -> b -> f a
adjuncted :: (Adjunction f g, Profunctor p, Functor h) => p (a -> g b) (h (c -> g d)) -> p (b -> f a) (h (d -> f c))
contrarepAdjunction :: Adjunction f g => (a -> f ()) -> g a
coindexAdjunction :: Adjunction f g => g a -> a -> f ()

Documentation

class (Contravariant f, Representable g) => Adjunction (f :: Type -> Type) (g :: Type -> Type) | f -> g, g -> f where Source #

An adjunction from Hask^op to Hask

Op (f a) b ~ Hask a (g b)

rightAdjunct unit = id
leftAdjunct counit = id

Any adjunction from Hask to Hask^op would indirectly permit unsafePerformIO, and therefore does not exist.

Minimal complete definition

Methods

unit :: a -> g (f a) Source #

counit :: a -> f (g a) Source #

leftAdjunct :: (b -> f a) -> a -> g b Source #

rightAdjunct :: (a -> g b) -> b -> f a Source #

Instances details

Adjunction Predicate Predicate Source #	This gives rise to the `Cont Bool` `Monad`
Instance details Defined in Data.Functor.Contravariant.Adjunction Methods unit :: a -> Predicate (Predicate a) Source # counit :: a -> Predicate (Predicate a) Source # leftAdjunct :: (b -> Predicate a) -> a -> Predicate b Source # rightAdjunct :: (a -> Predicate b) -> b -> Predicate a Source #
Adjunction (Op r) (Op r) Source #	This `Adjunction` gives rise to the `Cont` `Monad`
Instance details Defined in Data.Functor.Contravariant.Adjunction Methods unit :: a -> Op r (Op r a) Source # counit :: a -> Op r (Op r a) Source # leftAdjunct :: (b -> Op r a) -> a -> Op r b Source # rightAdjunct :: (a -> Op r b) -> b -> Op r a Source #

adjuncted :: (Adjunction f g, Profunctor p, Functor h) => p (a -> g b) (h (c -> g d)) -> p (b -> f a) (h (d -> f c)) Source #

leftAdjunct and rightAdjunct form two halves of an isomorphism.

This can be used with the combinators from the lens package.

adjuncted :: Adjunction f g => Iso' (b -> f a) (a -> g b)

Represent a Contravariant functor that has a left adjoint