profunctors-5.0.0.1: 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:

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` `id` ≡ `id``

If you supply `lmap` and `rmap`, ensure:

````lmap` `id` ≡ `id`
`rmap` `id` ≡ `id`
```

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.

``rmap` ≡ `dimap` `id``

Instances

 Profunctor (->) Monad m => Profunctor (Kleisli m) Profunctor (Tagged *) Functor w => Profunctor (Cokleisli w) Profunctor (Forget r) Arrow p => Profunctor (WrappedArrow p) Functor f => Profunctor (Costar f) Functor f => Profunctor (Star 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 `Star` 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

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) `Arrow` is `Strong` `Category` Functor m => Strong (Star 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 `Costar` 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) Choice (Tagged *) Comonad w => Choice (Cokleisli w) `extract` approximates `costrength` Monoid r => Choice (Forget r) ArrowChoice p => Choice (WrappedArrow p) Traversable w => Choice (Costar w) Applicative f => Choice (Star 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`

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 Star f d c Source

Lift a `Functor` into a `Profunctor` (forwards).

Constructors

 Star FieldsrunStar :: d -> f c

Instances

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

newtype Costar f d c Source

Lift a `Functor` into a `Profunctor` (backwards).

Constructors

 Costar FieldsrunCostar :: f d -> c

Instances

 Functor f => Profunctor (Costar f) Traversable w => Choice (Costar w) Functor f => Closed (Costar f) Functor f => Corepresentable (Costar f) Functor f => Cosieve (Costar f) f Monad (Costar f a) Functor (Costar f a) Applicative (Costar f a) Distributive (Costar f d) type Corep (Costar 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) `Arrow` is `Strong` `Category`

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) Sieve (Forget r) (Const 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