profunctors-4.4: Profunctors

Copyright(C) 2011-2013 Edward Kmett,
LicenseBSD-style (see the file LICENSE)
MaintainerEdward Kmett <ekmett@gmail.com>
Stabilityprovisional
Portabilityportable
Safe HaskellTrustworthy
LanguageHaskell2010

Data.Profunctor

Contents

Description

For a good explanation of profunctors in Haskell see Dan Piponi's article:

http://blog.sigfpe.com/2011/07/profunctors-in-haskell.html

For more information on strength and costrength, see:

http://comonad.com/reader/2008/deriving-strength-from-laziness/

Synopsis

Profunctors

class Profunctor p where Source

Formally, the class Profunctor represents a profunctor from Hask -> Hask.

Intuitively it is a bifunctor where the first argument is contravariant and the second argument is covariant.

You can define a Profunctor by either defining dimap or by defining both lmap and rmap.

If you supply dimap, you should ensure that:

dimap id idid

If you supply lmap and rmap, ensure:

lmap idid
rmap idid

If you supply both, you should also ensure:

dimap f g ≡ lmap f . rmap g

These ensure by parametricity:

dimap (f . g) (h . i) ≡ dimap g h . dimap f i
lmap (f . g) ≡ lmap g . lmap f
rmap (f . g) ≡ rmap f . rmap g

Minimal complete definition

dimap | lmap, rmap

Methods

dimap :: (a -> b) -> (c -> d) -> p b c -> p a d Source

Map over both arguments at the same time.

dimap f g ≡ lmap f . rmap g

lmap :: (a -> b) -> p b c -> p a c Source

Map the first argument contravariantly.

lmap f ≡ dimap f id

rmap :: (b -> c) -> p a b -> p a c Source

Map the second argument covariantly.

rmapdimap id

Profunctorial Strength

class Profunctor p => Strong p where Source

Generalizing UpStar of a strong Functor

Note: Every Functor in Haskell is strong with respect to (,).

This describes profunctor strength with respect to the product structure of Hask.

http://www-kb.is.s.u-tokyo.ac.jp/~asada/papers/arrStrMnd.pdf

Minimal complete definition

first' | second'

Methods

first' :: p a b -> p (a, c) (b, c) Source

second' :: p a b -> p (c, a) (c, b) Source

Instances

Strong (->) 
Monad m => Strong (Kleisli m) 
Strong (Forget r) 
Arrow p => Strong (WrappedArrow p)

Every Arrow is a Strong Monad in Prof

Functor m => Strong (UpStar m) 
Strong p => Strong (Closure p) 
Profunctor p => Strong (Tambara p) 
(Functor f, Strong p) => Strong (Cayley f p) 
(Strong p, Strong q) => Strong (Procompose p q) 

class Profunctor p => Choice p where Source

The generalization of DownStar of Functor that is strong with respect to Either.

Note: This is also a notion of strength, except with regards to another monoidal structure that we can choose to equip Hask with: the cocartesian coproduct.

Minimal complete definition

left' | right'

Methods

left' :: p a b -> p (Either a c) (Either b c) Source

right' :: p a b -> p (Either c a) (Either c b) Source

Instances

Choice (->) 
Monad m => Choice (Kleisli m) 
Comonad w => Choice (Cokleisli w)

extract approximates costrength

Choice (Tagged *) 
Monoid r => Choice (Forget r) 
ArrowChoice p => Choice (WrappedArrow p) 
Traversable w => Choice (DownStar w) 
Applicative f => Choice (UpStar f) 
Profunctor p => Choice (Cotambara p) 
Choice p => Choice (Tambara p) 
(Functor f, Choice p) => Choice (Cayley f p) 
(Choice p, Choice q) => Choice (Procompose p q) 

Profunctorial Costrength

class Profunctor p => Costrong p where Source

Analogous to ArrowLoop, loop = unfirst

unfirst . unfirst =

Minimal complete definition

Nothing

Methods

unfirst :: p (a, d) (b, d) -> p a b Source

unsecond :: p (d, a) (d, b) -> p a b Source

class Profunctor p => Cochoice p where Source

Minimal complete definition

Nothing

Methods

unleft :: p (Either a d) (Either b d) -> p a b Source

unright :: p (Either d a) (Either d b) -> p a b Source

Common Profunctors

newtype UpStar f d c Source

Lift a Functor into a Profunctor (forwards).

Constructors

UpStar 

Fields

runUpStar :: d -> f c
 

Instances

newtype DownStar f d c Source

Lift a Functor into a Profunctor (backwards).

Constructors

DownStar 

Fields

runDownStar :: f d -> c
 

newtype WrappedArrow p a b Source

Wrap an arrow for use as a Profunctor.

Constructors

WrapArrow 

Fields

unwrapArrow :: p a b
 

newtype Forget r a b Source

Constructors

Forget 

Fields

runForget :: a -> r
 

Instances

type (:->) p q = forall a b. p a b -> q a b infixr 0 Source