profunctors-4.4.1: Profunctors

Copyright (C) 2011-2013 Edward Kmett, BSD-style (see the file LICENSE) Edward Kmett provisional portable Trustworthy Haskell2010

Data.Profunctor

Description

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

For more information on strength and costrength, see:

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

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

Instances

 Profunctor (->) Monad m => Profunctor (Kleisli m) Functor w => Profunctor (Cokleisli w) Profunctor (Tagged *) Profunctor (Forget r) Arrow p => Profunctor (WrappedArrow p) Functor f => Profunctor (DownStar f) Functor f => Profunctor (UpStar f) Profunctor p => Profunctor (Environment p) Profunctor p => Profunctor (Closure p) Profunctor p => Profunctor (Codensity p) Profunctor p => Profunctor (Copastro p) Profunctor p => Profunctor (Cotambara p) Profunctor p => Profunctor (Pastro p) Profunctor p => Profunctor (Tambara p) (Functor f, Profunctor p) => Profunctor (Cayley f p) (Profunctor p, Profunctor q) => Profunctor (Rift p q) (Profunctor p, Profunctor q) => Profunctor (Procompose p q) (Profunctor p, Profunctor q) => Profunctor (Ran p q)

## 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.

Minimal complete definition

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

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

Instances

 Costrong (->) MonadFix m => Costrong (Kleisli m) Costrong (Tagged *) ArrowLoop p => Costrong (WrappedArrow p)

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 FieldsrunUpStar :: d -> f c

Instances

 Functor f => Profunctor (UpStar f) Applicative f => Choice (UpStar f) Functor m => Strong (UpStar m) Distributive f => Closed (UpStar f) Functor f => Representable (UpStar f) Alternative f => Alternative (UpStar f a) Monad f => Monad (UpStar f a) Functor f => Functor (UpStar f a) MonadPlus f => MonadPlus (UpStar f a) Applicative f => Applicative (UpStar f a) type Rep (UpStar f) = f

newtype DownStar f d c Source

Lift a Functor into a Profunctor (backwards).

Constructors

 DownStar FieldsrunDownStar :: f d -> c

Instances

 Functor f => Profunctor (DownStar f) Traversable w => Choice (DownStar w) Functor f => Closed (DownStar f) Functor f => Corepresentable (DownStar f) Monad (DownStar f a) Functor (DownStar f a) Applicative (DownStar f a) type Corep (DownStar f) = f

newtype WrappedArrow p a b Source

Wrap an arrow for use as a Profunctor.

Constructors

 WrapArrow FieldsunwrapArrow :: p a b

Instances

 Category * p => Category * (WrappedArrow p) Arrow p => Arrow (WrappedArrow p) ArrowZero p => ArrowZero (WrappedArrow p) ArrowChoice p => ArrowChoice (WrappedArrow p) ArrowApply p => ArrowApply (WrappedArrow p) ArrowLoop p => ArrowLoop (WrappedArrow p) Arrow p => Profunctor (WrappedArrow p) ArrowLoop p => Costrong (WrappedArrow p) ArrowChoice p => Choice (WrappedArrow p) Arrow p => Strong (WrappedArrow p) Every Arrow is a Strong Monad in Prof

newtype Forget r a b Source

Constructors

 Forget FieldsrunForget :: a -> r

Instances

 Profunctor (Forget r) Monoid r => Choice (Forget r) Strong (Forget r) Representable (Forget r) Functor (Forget r a) Foldable (Forget r a) Traversable (Forget r a) type Rep (Forget r) = Const r

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