profunctors-5.2: Profunctors

Copyright | (C) 2011-2015 Edward Kmett,
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | portable
Safe Haskell | Trustworthy
Language | Haskell2010

Data.Profunctor

Contents

- Profunctors
  - Profunctorial Strength
  - Closed
  - Profunctorial Costrength
  - Common Profunctors

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

- class Profunctor p where
- class Profunctor p => Strong p where
- uncurry' :: Strong p => p a (b -> c) -> p (a, b) c
- class Profunctor p => Choice p where
- class Profunctor p => Closed p where
- curry' :: Closed p => p (a, b) c -> p a (b -> c)
- class (Traversing p, Closed p) => Mapping p where
- class Profunctor p => Costrong p where
- class Profunctor p => Cochoice p where
- newtype Star f d c = Star {
  - runStar :: d -> f c
  }
- newtype Costar f d c = Costar {
  - runCostar :: f d -> c
  }
- newtype WrappedArrow p a b = WrapArrow {
  - unwrapArrow :: p a b
  }
- newtype Forget r a b = Forget {
  - runForget :: a -> r
  }
- type (:->) p q = forall a b. p a b -> q a b

# Profunctors

class Profunctor p where #

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

dimap | lmap, rmap

Methods

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

Map over both arguments at the same time.

    dimap f g ≡ lmap f . rmap g

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

Map the first argument contravariantly.

    lmap f ≡ dimap f id

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

Map the second argument covariantly.

    rmap ≡ dimap id

Instances

Profunctor (->) # |
Methods dimap :: (a -> b) -> (c -> d) -> (b -> c) -> a -> d # lmap :: (a -> b) -> (b -> c) -> a -> c # rmap :: (b -> c) -> (a -> b) -> a -> c # (#.) :: Coercible * c b => (b -> c) -> (a -> b) -> a -> c # (.#) :: Coercible * b a => (b -> c) -> (a -> b) -> a -> c #
Monad m => Profunctor (Kleisli m) # |
Methods dimap :: (a -> b) -> (c -> d) -> Kleisli m b c -> Kleisli m a d # lmap :: (a -> b) -> Kleisli m b c -> Kleisli m a c # rmap :: (b -> c) -> Kleisli m a b -> Kleisli m a c # (#.) :: Coercible * c b => (b -> c) -> Kleisli m a b -> Kleisli m a c # (.#) :: Coercible * b a => Kleisli m b c -> (a -> b) -> Kleisli m a c #
Functor w => Profunctor (Cokleisli w) # |
Methods dimap :: (a -> b) -> (c -> d) -> Cokleisli w b c -> Cokleisli w a d # lmap :: (a -> b) -> Cokleisli w b c -> Cokleisli w a c # rmap :: (b -> c) -> Cokleisli w a b -> Cokleisli w a c # (#.) :: Coercible * c b => (b -> c) -> Cokleisli w a b -> Cokleisli w a c # (.#) :: Coercible * b a => Cokleisli w b c -> (a -> b) -> Cokleisli w a c #
Profunctor (Tagged *) # |
Methods dimap :: (a -> b) -> (c -> d) -> Tagged * b c -> Tagged * a d # lmap :: (a -> b) -> Tagged * b c -> Tagged * a c # rmap :: (b -> c) -> Tagged * a b -> Tagged * a c # (#.) :: Coercible * c b => (b -> c) -> Tagged * a b -> Tagged * a c # (.#) :: Coercible * b a => Tagged * b c -> (a -> b) -> Tagged * a c #
Profunctor (Forget r) # |
Methods dimap :: (a -> b) -> (c -> d) -> Forget r b c -> Forget r a d # lmap :: (a -> b) -> Forget r b c -> Forget r a c # rmap :: (b -> c) -> Forget r a b -> Forget r a c # (#.) :: Coercible * c b => (b -> c) -> Forget r a b -> Forget r a c # (.#) :: Coercible * b a => Forget r b c -> (a -> b) -> Forget r a c #
Arrow p => Profunctor (WrappedArrow p) # |
Methods dimap :: (a -> b) -> (c -> d) -> WrappedArrow p b c -> WrappedArrow p a d # lmap :: (a -> b) -> WrappedArrow p b c -> WrappedArrow p a c # rmap :: (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (#.) :: Coercible * c b => (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (.#) :: Coercible * b a => WrappedArrow p b c -> (a -> b) -> WrappedArrow p a c #
Functor f => Profunctor (Costar f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Costar f b c -> Costar f a d # lmap :: (a -> b) -> Costar f b c -> Costar f a c # rmap :: (b -> c) -> Costar f a b -> Costar f a c # (#.) :: Coercible * c b => (b -> c) -> Costar f a b -> Costar f a c # (.#) :: Coercible * b a => Costar f b c -> (a -> b) -> Costar f a c #
Functor f => Profunctor (Star f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Star f b c -> Star f a d # lmap :: (a -> b) -> Star f b c -> Star f a c # rmap :: (b -> c) -> Star f a b -> Star f a c # (#.) :: Coercible * c b => (b -> c) -> Star f a b -> Star f a c # (.#) :: Coercible * b a => Star f b c -> (a -> b) -> Star f a c #
Profunctor (Copastro p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Copastro p b c -> Copastro p a d # lmap :: (a -> b) -> Copastro p b c -> Copastro p a c # rmap :: (b -> c) -> Copastro p a b -> Copastro p a c # (#.) :: Coercible * c b => (b -> c) -> Copastro p a b -> Copastro p a c # (.#) :: Coercible * b a => Copastro p b c -> (a -> b) -> Copastro p a c #
Profunctor (Cotambara p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Cotambara p b c -> Cotambara p a d # lmap :: (a -> b) -> Cotambara p b c -> Cotambara p a c # rmap :: (b -> c) -> Cotambara p a b -> Cotambara p a c # (#.) :: Coercible * c b => (b -> c) -> Cotambara p a b -> Cotambara p a c # (.#) :: Coercible * b a => Cotambara p b c -> (a -> b) -> Cotambara p a c #
Profunctor (Pastro p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Pastro p b c -> Pastro p a d # lmap :: (a -> b) -> Pastro p b c -> Pastro p a c # rmap :: (b -> c) -> Pastro p a b -> Pastro p a c # (#.) :: Coercible * c b => (b -> c) -> Pastro p a b -> Pastro p a c # (.#) :: Coercible * b a => Pastro p b c -> (a -> b) -> Pastro p a c #
Profunctor p => Profunctor (Tambara p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Tambara p b c -> Tambara p a d # lmap :: (a -> b) -> Tambara p b c -> Tambara p a c # rmap :: (b -> c) -> Tambara p a b -> Tambara p a c # (#.) :: Coercible * c b => (b -> c) -> Tambara p a b -> Tambara p a c # (.#) :: Coercible * b a => Tambara p b c -> (a -> b) -> Tambara p a c #
Profunctor (Environment p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Environment p b c -> Environment p a d # lmap :: (a -> b) -> Environment p b c -> Environment p a c # rmap :: (b -> c) -> Environment p a b -> Environment p a c # (#.) :: Coercible * c b => (b -> c) -> Environment p a b -> Environment p a c # (.#) :: Coercible * b a => Environment p b c -> (a -> b) -> Environment p a c #
Profunctor p => Profunctor (Closure p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Closure p b c -> Closure p a d # lmap :: (a -> b) -> Closure p b c -> Closure p a c # rmap :: (b -> c) -> Closure p a b -> Closure p a c # (#.) :: Coercible * c b => (b -> c) -> Closure p a b -> Closure p a c # (.#) :: Coercible * b a => Closure p b c -> (a -> b) -> Closure p a c #
Profunctor (CopastroSum p) # |
Methods dimap :: (a -> b) -> (c -> d) -> CopastroSum p b c -> CopastroSum p a d # lmap :: (a -> b) -> CopastroSum p b c -> CopastroSum p a c # rmap :: (b -> c) -> CopastroSum p a b -> CopastroSum p a c # (#.) :: Coercible * c b => (b -> c) -> CopastroSum p a b -> CopastroSum p a c # (.#) :: Coercible * b a => CopastroSum p b c -> (a -> b) -> CopastroSum p a c #
Profunctor (CotambaraSum p) # |
Methods dimap :: (a -> b) -> (c -> d) -> CotambaraSum p b c -> CotambaraSum p a d # lmap :: (a -> b) -> CotambaraSum p b c -> CotambaraSum p a c # rmap :: (b -> c) -> CotambaraSum p a b -> CotambaraSum p a c # (#.) :: Coercible * c b => (b -> c) -> CotambaraSum p a b -> CotambaraSum p a c # (.#) :: Coercible * b a => CotambaraSum p b c -> (a -> b) -> CotambaraSum p a c #
Profunctor (PastroSum p) # |
Methods dimap :: (a -> b) -> (c -> d) -> PastroSum p b c -> PastroSum p a d # lmap :: (a -> b) -> PastroSum p b c -> PastroSum p a c # rmap :: (b -> c) -> PastroSum p a b -> PastroSum p a c # (#.) :: Coercible * c b => (b -> c) -> PastroSum p a b -> PastroSum p a c # (.#) :: Coercible * b a => PastroSum p b c -> (a -> b) -> PastroSum p a c #
Profunctor p => Profunctor (TambaraSum p) # |
Methods dimap :: (a -> b) -> (c -> d) -> TambaraSum p b c -> TambaraSum p a d # lmap :: (a -> b) -> TambaraSum p b c -> TambaraSum p a c # rmap :: (b -> c) -> TambaraSum p a b -> TambaraSum p a c # (#.) :: Coercible * c b => (b -> c) -> TambaraSum p a b -> TambaraSum p a c # (.#) :: Coercible * b a => TambaraSum p b c -> (a -> b) -> TambaraSum p a c #
Profunctor (FreeTraversing p) # |
Methods dimap :: (a -> b) -> (c -> d) -> FreeTraversing p b c -> FreeTraversing p a d # lmap :: (a -> b) -> FreeTraversing p b c -> FreeTraversing p a c # rmap :: (b -> c) -> FreeTraversing p a b -> FreeTraversing p a c # (#.) :: Coercible * c b => (b -> c) -> FreeTraversing p a b -> FreeTraversing p a c # (.#) :: Coercible * b a => FreeTraversing p b c -> (a -> b) -> FreeTraversing p a c #
Profunctor p => Profunctor (CofreeTraversing p) # |
Methods dimap :: (a -> b) -> (c -> d) -> CofreeTraversing p b c -> CofreeTraversing p a d # lmap :: (a -> b) -> CofreeTraversing p b c -> CofreeTraversing p a c # rmap :: (b -> c) -> CofreeTraversing p a b -> CofreeTraversing p a c # (#.) :: Coercible * c b => (b -> c) -> CofreeTraversing p a b -> CofreeTraversing p a c # (.#) :: Coercible * b a => CofreeTraversing p b c -> (a -> b) -> CofreeTraversing p a c #
Profunctor (FreeMapping p) # |
Methods dimap :: (a -> b) -> (c -> d) -> FreeMapping p b c -> FreeMapping p a d # lmap :: (a -> b) -> FreeMapping p b c -> FreeMapping p a c # rmap :: (b -> c) -> FreeMapping p a b -> FreeMapping p a c # (#.) :: Coercible * c b => (b -> c) -> FreeMapping p a b -> FreeMapping p a c # (.#) :: Coercible * b a => FreeMapping p b c -> (a -> b) -> FreeMapping p a c #
Profunctor p => Profunctor (CofreeMapping p) # |
Methods dimap :: (a -> b) -> (c -> d) -> CofreeMapping p b c -> CofreeMapping p a d # lmap :: (a -> b) -> CofreeMapping p b c -> CofreeMapping p a c # rmap :: (b -> c) -> CofreeMapping p a b -> CofreeMapping p a c # (#.) :: Coercible * c b => (b -> c) -> CofreeMapping p a b -> CofreeMapping p a c # (.#) :: Coercible * b a => CofreeMapping p b c -> (a -> b) -> CofreeMapping p a c #
Profunctor p => Profunctor (Codensity p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Codensity p b c -> Codensity p a d # lmap :: (a -> b) -> Codensity p b c -> Codensity p a c # rmap :: (b -> c) -> Codensity p a b -> Codensity p a c # (#.) :: Coercible * c b => (b -> c) -> Codensity p a b -> Codensity p a c # (.#) :: Coercible * b a => Codensity p b c -> (a -> b) -> Codensity p a c #
(Functor f, Profunctor p) => Profunctor (Cayley f p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Cayley f p b c -> Cayley f p a d # lmap :: (a -> b) -> Cayley f p b c -> Cayley f p a c # rmap :: (b -> c) -> Cayley f p a b -> Cayley f p a c # (#.) :: Coercible * c b => (b -> c) -> Cayley f p a b -> Cayley f p a c # (.#) :: Coercible * b a => Cayley f p b c -> (a -> b) -> Cayley f p a c #
(Profunctor p, Profunctor q) => Profunctor (Rift p q) # |
Methods dimap :: (a -> b) -> (c -> d) -> Rift p q b c -> Rift p q a d # lmap :: (a -> b) -> Rift p q b c -> Rift p q a c # rmap :: (b -> c) -> Rift p q a b -> Rift p q a c # (#.) :: Coercible * c b => (b -> c) -> Rift p q a b -> Rift p q a c # (.#) :: Coercible * b a => Rift p q b c -> (a -> b) -> Rift p q a c #
(Profunctor p, Profunctor q) => Profunctor (Procompose p q) # |
Methods dimap :: (a -> b) -> (c -> d) -> Procompose p q b c -> Procompose p q a d # lmap :: (a -> b) -> Procompose p q b c -> Procompose p q a c # rmap :: (b -> c) -> Procompose p q a b -> Procompose p q a c # (#.) :: Coercible * c b => (b -> c) -> Procompose p q a b -> Procompose p q a c # (.#) :: Coercible * b a => Procompose p q b c -> (a -> b) -> Procompose p q a c #
(Profunctor p, Profunctor q) => Profunctor (Ran p q) # |
Methods dimap :: (a -> b) -> (c -> d) -> Ran p q b c -> Ran p q a d # lmap :: (a -> b) -> Ran p q b c -> Ran p q a c # rmap :: (b -> c) -> Ran p q a b -> Ran p q a c # (#.) :: Coercible * c b => (b -> c) -> Ran p q a b -> Ran p q a c # (.#) :: Coercible * b a => Ran p q b c -> (a -> b) -> Ran p q a c #
Functor f => Profunctor (Joker * * f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Joker * * f b c -> Joker * * f a d # lmap :: (a -> b) -> Joker * * f b c -> Joker * * f a c # rmap :: (b -> c) -> Joker * * f a b -> Joker * * f a c # (#.) :: Coercible * c b => (b -> c) -> Joker * * f a b -> Joker * * f a c # (.#) :: Coercible * b a => Joker * * f b c -> (a -> b) -> Joker * * f a c #
Contravariant f => Profunctor (Clown * * f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Clown * * f b c -> Clown * * f a d # lmap :: (a -> b) -> Clown * * f b c -> Clown * * f a c # rmap :: (b -> c) -> Clown * * f a b -> Clown * * f a c # (#.) :: Coercible * c b => (b -> c) -> Clown * * f a b -> Clown * * f a c # (.#) :: Coercible * b a => Clown * * f b c -> (a -> b) -> Clown * * f a c #
(Profunctor p, Profunctor q) => Profunctor (Product * * p q) # |
Methods dimap :: (a -> b) -> (c -> d) -> Product * * p q b c -> Product * * p q a d # lmap :: (a -> b) -> Product * * p q b c -> Product * * p q a c # rmap :: (b -> c) -> Product * * p q a b -> Product * * p q a c # (#.) :: Coercible * c b => (b -> c) -> Product * * p q a b -> Product * * p q a c # (.#) :: Coercible * b a => Product * * p q b c -> (a -> b) -> Product * * p q a c #
(Functor f, Profunctor p) => Profunctor (Tannen * * * f p) # |
Methods dimap :: (a -> b) -> (c -> d) -> Tannen * * * f p b c -> Tannen * * * f p a d # lmap :: (a -> b) -> Tannen * * * f p b c -> Tannen * * * f p a c # rmap :: (b -> c) -> Tannen * * * f p a b -> Tannen * * * f p a c # (#.) :: Coercible * c b => (b -> c) -> Tannen * * * f p a b -> Tannen * * * f p a c # (.#) :: Coercible * b a => Tannen * * * f p b c -> (a -> b) -> Tannen * * * f p a c #
(Profunctor p, Functor f, Functor g) => Profunctor (Biff * * * * p f g) # |
Methods dimap :: (a -> b) -> (c -> d) -> Biff * * * * p f g b c -> Biff * * * * p f g a d # lmap :: (a -> b) -> Biff * * * * p f g b c -> Biff * * * * p f g a c # rmap :: (b -> c) -> Biff * * * * p f g a b -> Biff * * * * p f g a c # (#.) :: Coercible * c b => (b -> c) -> Biff * * * * p f g a b -> Biff * * * * p f g a c # (.#) :: Coercible * b a => Biff * * * * p f g b c -> (a -> b) -> Biff * * * * p f g a c #

## Profunctorial Strength

class Profunctor p => Strong p where #

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

first' | second'

Methods

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

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

Instances

Strong (->) # |
Methods first' :: (a -> b) -> (a, c) -> (b, c) # second' :: (a -> b) -> (c, a) -> (c, b) #
Monad m => Strong (Kleisli m) # |
Methods first' :: Kleisli m a b -> Kleisli m (a, c) (b, c) # second' :: Kleisli m a b -> Kleisli m (c, a) (c, b) #
Strong (Forget r) # |
Methods first' :: Forget r a b -> Forget r (a, c) (b, c) # second' :: Forget r a b -> Forget r (c, a) (c, b) #
Arrow p => Strong (WrappedArrow p) # | `Arrow` is `Strong` `Category`
Methods first' :: WrappedArrow p a b -> WrappedArrow p (a, c) (b, c) # second' :: WrappedArrow p a b -> WrappedArrow p (c, a) (c, b) #
Functor m => Strong (Star m) # |
Methods first' :: Star m a b -> Star m (a, c) (b, c) # second' :: Star m a b -> Star m (c, a) (c, b) #
Strong (Pastro p) # |
Methods first' :: Pastro p a b -> Pastro p (a, c) (b, c) # second' :: Pastro p a b -> Pastro p (c, a) (c, b) #
Profunctor p => Strong (Tambara p) # |
Methods first' :: Tambara p a b -> Tambara p (a, c) (b, c) # second' :: Tambara p a b -> Tambara p (c, a) (c, b) #
Strong p => Strong (Closure p) # |
Methods first' :: Closure p a b -> Closure p (a, c) (b, c) # second' :: Closure p a b -> Closure p (c, a) (c, b) #
Strong (FreeTraversing p) # |
Methods first' :: FreeTraversing p a b -> FreeTraversing p (a, c) (b, c) # second' :: FreeTraversing p a b -> FreeTraversing p (c, a) (c, b) #
Profunctor p => Strong (CofreeTraversing p) # |
Methods first' :: CofreeTraversing p a b -> CofreeTraversing p (a, c) (b, c) # second' :: CofreeTraversing p a b -> CofreeTraversing p (c, a) (c, b) #
Strong (FreeMapping p) # |
Methods first' :: FreeMapping p a b -> FreeMapping p (a, c) (b, c) # second' :: FreeMapping p a b -> FreeMapping p (c, a) (c, b) #
Profunctor p => Strong (CofreeMapping p) # |
Methods first' :: CofreeMapping p a b -> CofreeMapping p (a, c) (b, c) # second' :: CofreeMapping p a b -> CofreeMapping p (c, a) (c, b) #
(Functor f, Strong p) => Strong (Cayley f p) # |
Methods first' :: Cayley f p a b -> Cayley f p (a, c) (b, c) # second' :: Cayley f p a b -> Cayley f p (c, a) (c, b) #
(Strong p, Strong q) => Strong (Procompose p q) # |
Methods first' :: Procompose p q a b -> Procompose p q (a, c) (b, c) # second' :: Procompose p q a b -> Procompose p q (c, a) (c, b) #
Contravariant f => Strong (Clown * * f) # |
Methods first' :: Clown * * f a b -> Clown * * f (a, c) (b, c) # second' :: Clown * * f a b -> Clown * * f (c, a) (c, b) #
(Strong p, Strong q) => Strong (Product * * p q) # |
Methods first' :: Product * * p q a b -> Product * * p q (a, c) (b, c) # second' :: Product * * p q a b -> Product * * p q (c, a) (c, b) #
(Functor f, Strong p) => Strong (Tannen * * * f p) # |
Methods first' :: Tannen * * * f p a b -> Tannen * * * f p (a, c) (b, c) # second' :: Tannen * * * f p a b -> Tannen * * * f p (c, a) (c, b) #

uncurry' :: Strong p => p a (b -> c) -> p (a, b) c #

class Profunctor p => Choice p where #

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

left' | right'

Methods

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

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

Instances

Choice (->) # |
Methods left' :: (a -> b) -> Either a c -> Either b c # right' :: (a -> b) -> Either c a -> Either c b #
Monad m => Choice (Kleisli m) # |
Methods left' :: Kleisli m a b -> Kleisli m (Either a c) (Either b c) # right' :: Kleisli m a b -> Kleisli m (Either c a) (Either c b) #
Comonad w => Choice (Cokleisli w) # | `extract` approximates `costrength`
Methods left' :: Cokleisli w a b -> Cokleisli w (Either a c) (Either b c) # right' :: Cokleisli w a b -> Cokleisli w (Either c a) (Either c b) #
Choice (Tagged *) # |
Methods left' :: Tagged * a b -> Tagged * (Either a c) (Either b c) # right' :: Tagged * a b -> Tagged * (Either c a) (Either c b) #
Monoid r => Choice (Forget r) # |
Methods left' :: Forget r a b -> Forget r (Either a c) (Either b c) # right' :: Forget r a b -> Forget r (Either c a) (Either c b) #
ArrowChoice p => Choice (WrappedArrow p) # |
Methods left' :: WrappedArrow p a b -> WrappedArrow p (Either a c) (Either b c) # right' :: WrappedArrow p a b -> WrappedArrow p (Either c a) (Either c b) #
Traversable w => Choice (Costar w) # |
Methods left' :: Costar w a b -> Costar w (Either a c) (Either b c) # right' :: Costar w a b -> Costar w (Either c a) (Either c b) #
Applicative f => Choice (Star f) # |
Methods left' :: Star f a b -> Star f (Either a c) (Either b c) # right' :: Star f a b -> Star f (Either c a) (Either c b) #
Choice p => Choice (Tambara p) # |
Methods left' :: Tambara p a b -> Tambara p (Either a c) (Either b c) # right' :: Tambara p a b -> Tambara p (Either c a) (Either c b) #
Choice (PastroSum p) # |
Methods left' :: PastroSum p a b -> PastroSum p (Either a c) (Either b c) # right' :: PastroSum p a b -> PastroSum p (Either c a) (Either c b) #
Profunctor p => Choice (TambaraSum p) # |
Methods left' :: TambaraSum p a b -> TambaraSum p (Either a c) (Either b c) # right' :: TambaraSum p a b -> TambaraSum p (Either c a) (Either c b) #
Choice (FreeTraversing p) # |
Methods left' :: FreeTraversing p a b -> FreeTraversing p (Either a c) (Either b c) # right' :: FreeTraversing p a b -> FreeTraversing p (Either c a) (Either c b) #
Profunctor p => Choice (CofreeTraversing p) # |
Methods left' :: CofreeTraversing p a b -> CofreeTraversing p (Either a c) (Either b c) # right' :: CofreeTraversing p a b -> CofreeTraversing p (Either c a) (Either c b) #
Choice (FreeMapping p) # |
Methods left' :: FreeMapping p a b -> FreeMapping p (Either a c) (Either b c) # right' :: FreeMapping p a b -> FreeMapping p (Either c a) (Either c b) #
Profunctor p => Choice (CofreeMapping p) # |
Methods left' :: CofreeMapping p a b -> CofreeMapping p (Either a c) (Either b c) # right' :: CofreeMapping p a b -> CofreeMapping p (Either c a) (Either c b) #
(Functor f, Choice p) => Choice (Cayley f p) # |
Methods left' :: Cayley f p a b -> Cayley f p (Either a c) (Either b c) # right' :: Cayley f p a b -> Cayley f p (Either c a) (Either c b) #
(Choice p, Choice q) => Choice (Procompose p q) # |
Methods left' :: Procompose p q a b -> Procompose p q (Either a c) (Either b c) # right' :: Procompose p q a b -> Procompose p q (Either c a) (Either c b) #
Functor f => Choice (Joker * * f) # |
Methods left' :: Joker * * f a b -> Joker * * f (Either a c) (Either b c) # right' :: Joker * * f a b -> Joker * * f (Either c a) (Either c b) #
(Choice p, Choice q) => Choice (Product * * p q) # |
Methods left' :: Product * * p q a b -> Product * * p q (Either a c) (Either b c) # right' :: Product * * p q a b -> Product * * p q (Either c a) (Either c b) #
(Functor f, Choice p) => Choice (Tannen * * * f p) # |
Methods left' :: Tannen * * * f p a b -> Tannen * * * f p (Either a c) (Either b c) # right' :: Tannen * * * f p a b -> Tannen * * * f p (Either c a) (Either c b) #

## Closed

class Profunctor p => Closed p where #

A strong profunctor allows the monoidal structure to pass through.

A closed profunctor allows the closed structure to pass through.

Minimal complete definition

closed

Methods

closed :: p a b -> p (x -> a) (x -> b) #

Instances

Closed (->) # |
Methods closed :: (a -> b) -> (x -> a) -> x -> b #
(Distributive f, Monad f) => Closed (Kleisli f) # |
Methods closed :: Kleisli f a b -> Kleisli f (x -> a) (x -> b) #
Functor f => Closed (Cokleisli f) # |
Methods closed :: Cokleisli f a b -> Cokleisli f (x -> a) (x -> b) #
Closed (Tagged *) # |
Methods closed :: Tagged * a b -> Tagged * (x -> a) (x -> b) #
Functor f => Closed (Costar f) # |
Methods closed :: Costar f a b -> Costar f (x -> a) (x -> b) #
Distributive f => Closed (Star f) # |
Methods closed :: Star f a b -> Star f (x -> a) (x -> b) #
Closed (Environment p) # |
Methods closed :: Environment p a b -> Environment p (x -> a) (x -> b) #
Profunctor p => Closed (Closure p) # |
Methods closed :: Closure p a b -> Closure p (x -> a) (x -> b) #
Closed (FreeMapping p) # |
Methods closed :: FreeMapping p a b -> FreeMapping p (x -> a) (x -> b) #
Profunctor p => Closed (CofreeMapping p) # |
Methods closed :: CofreeMapping p a b -> CofreeMapping p (x -> a) (x -> b) #
(Closed p, Closed q) => Closed (Procompose p q) # |
Methods closed :: Procompose p q a b -> Procompose p q (x -> a) (x -> b) #
(Closed p, Closed q) => Closed (Product * * p q) # |
Methods closed :: Product * * p q a b -> Product * * p q (x -> a) (x -> b) #
(Functor f, Closed p) => Closed (Tannen * * * f p) # |
Methods closed :: Tannen * * * f p a b -> Tannen * * * f p (x -> a) (x -> b) #

curry' :: Closed p => p (a, b) c -> p a (b -> c) #

class (Traversing p, Closed p) => Mapping p where #

Minimal complete definition

map'

Methods

map' :: Functor f => p a b -> p (f a) (f b) #

Instances

Mapping (->) # |
Methods map' :: Functor f => (a -> b) -> f a -> f b #
(Monad m, Distributive m) => Mapping (Kleisli m) # |
Methods map' :: Functor f => Kleisli m a b -> Kleisli m (f a) (f b) #
(Applicative m, Distributive m) => Mapping (Star m) # |
Methods map' :: Functor f => Star m a b -> Star m (f a) (f b) #
Mapping (FreeMapping p) # |
Methods map' :: Functor f => FreeMapping p a b -> FreeMapping p (f a) (f b) #
Profunctor p => Mapping (CofreeMapping p) # |
Methods map' :: Functor f => CofreeMapping p a b -> CofreeMapping p (f a) (f b) #

## Profunctorial Costrength

class Profunctor p => Costrong p where #

Analogous to `ArrowLoop`, `loop` = `unfirst`

Minimal complete definition

unfirst | unsecond

Methods

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

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

Instances

Costrong (->) # |
Methods unfirst :: ((a, d) -> (b, d)) -> a -> b # unsecond :: ((d, a) -> (d, b)) -> a -> b #
MonadFix m => Costrong (Kleisli m) # |
Methods unfirst :: Kleisli m (a, d) (b, d) -> Kleisli m a b # unsecond :: Kleisli m (d, a) (d, b) -> Kleisli m a b #
Functor f => Costrong (Cokleisli f) # |
Methods unfirst :: Cokleisli f (a, d) (b, d) -> Cokleisli f a b # unsecond :: Cokleisli f (d, a) (d, b) -> Cokleisli f a b #
Costrong (Tagged *) # |
Methods unfirst :: Tagged * (a, d) (b, d) -> Tagged * a b # unsecond :: Tagged * (d, a) (d, b) -> Tagged * a b #
ArrowLoop p => Costrong (WrappedArrow p) # |
Methods unfirst :: WrappedArrow p (a, d) (b, d) -> WrappedArrow p a b # unsecond :: WrappedArrow p (d, a) (d, b) -> WrappedArrow p a b #
Functor f => Costrong (Costar f) # |
Methods unfirst :: Costar f (a, d) (b, d) -> Costar f a b # unsecond :: Costar f (d, a) (d, b) -> Costar f a b #
Costrong (Copastro p) # |
Methods unfirst :: Copastro p (a, d) (b, d) -> Copastro p a b # unsecond :: Copastro p (d, a) (d, b) -> Copastro p a b #
Costrong (Cotambara p) # |
Methods unfirst :: Cotambara p (a, d) (b, d) -> Cotambara p a b # unsecond :: Cotambara p (d, a) (d, b) -> Cotambara p a b #
(Corepresentable p, Corepresentable q) => Costrong (Procompose p q) # |
Methods unfirst :: Procompose p q (a, d) (b, d) -> Procompose p q a b # unsecond :: Procompose p q (d, a) (d, b) -> Procompose p q a b #
(Costrong p, Costrong q) => Costrong (Product * * p q) # |
Methods unfirst :: Product * * p q (a, d) (b, d) -> Product * * p q a b # unsecond :: Product * * p q (d, a) (d, b) -> Product * * p q a b #
(Functor f, Costrong p) => Costrong (Tannen * * * f p) # |
Methods unfirst :: Tannen * * * f p (a, d) (b, d) -> Tannen * * * f p a b # unsecond :: Tannen * * * f p (d, a) (d, b) -> Tannen * * * f p a b #

class Profunctor p => Cochoice p where #

Minimal complete definition

unleft | unright

Methods

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

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

Instances

Cochoice (->) # |
Methods unleft :: (Either a d -> Either b d) -> a -> b # unright :: (Either d a -> Either d b) -> a -> b #
Applicative f => Cochoice (Costar f) # |
Methods unleft :: Costar f (Either a d) (Either b d) -> Costar f a b # unright :: Costar f (Either d a) (Either d b) -> Costar f a b #
Traversable f => Cochoice (Star f) # |
Methods unleft :: Star f (Either a d) (Either b d) -> Star f a b # unright :: Star f (Either d a) (Either d b) -> Star f a b #
Cochoice (CopastroSum p) # |
Methods unleft :: CopastroSum p (Either a d) (Either b d) -> CopastroSum p a b # unright :: CopastroSum p (Either d a) (Either d b) -> CopastroSum p a b #
Cochoice (CotambaraSum p) # |
Methods unleft :: CotambaraSum p (Either a d) (Either b d) -> CotambaraSum p a b # unright :: CotambaraSum p (Either d a) (Either d b) -> CotambaraSum p a b #
(Cochoice p, Cochoice q) => Cochoice (Product * * p q) # |
Methods unleft :: Product * * p q (Either a d) (Either b d) -> Product * * p q a b # unright :: Product * * p q (Either d a) (Either d b) -> Product * * p q a b #
(Functor f, Cochoice p) => Cochoice (Tannen * * * f p) # |
Methods unleft :: Tannen * * * f p (Either a d) (Either b d) -> Tannen * * * f p a b # unright :: Tannen * * * f p (Either d a) (Either d b) -> Tannen * * * f p a b #

## Common Profunctors

newtype Star f d c #

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

Constructors

Star |
Fields runStar :: d -> f c

Instances

Functor f => Profunctor (Star f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Star f b c -> Star f a d # lmap :: (a -> b) -> Star f b c -> Star f a c # rmap :: (b -> c) -> Star f a b -> Star f a c # (#.) :: Coercible * c b => (b -> c) -> Star f a b -> Star f a c # (.#) :: Coercible * b a => Star f b c -> (a -> b) -> Star f a c #
Functor m => Strong (Star m) # |
Methods first' :: Star m a b -> Star m (a, c) (b, c) # second' :: Star m a b -> Star m (c, a) (c, b) #
Distributive f => Closed (Star f) # |
Methods closed :: Star f a b -> Star f (x -> a) (x -> b) #
Traversable f => Cochoice (Star f) # |
Methods unleft :: Star f (Either a d) (Either b d) -> Star f a b # unright :: Star f (Either d a) (Either d b) -> Star f a b #
Applicative f => Choice (Star f) # |
Methods left' :: Star f a b -> Star f (Either a c) (Either b c) # right' :: Star f a b -> Star f (Either c a) (Either c b) #
Applicative m => Traversing (Star m) # |
Methods traverse' :: Traversable f => Star m a b -> Star m (f a) (f b) # wander :: (forall f. Applicative f => (a -> f b) -> s -> f t) -> Star m a b -> Star m s t #
(Applicative m, Distributive m) => Mapping (Star m) # |
Methods map' :: Functor f => Star m a b -> Star m (f a) (f b) #
Functor f => Representable (Star f) # |
Associated Types type Rep (Star f :: * -> * -> *) :: * -> * # Methods tabulate :: (d -> Rep (Star f) c) -> Star f d c #
Functor f => Sieve (Star f) f # |
Methods sieve :: Star f a b -> a -> f b #
Monad f => Monad (Star f a) # |
Methods (>>=) :: Star f a a -> (a -> Star f a b) -> Star f a b # (>>) :: Star f a a -> Star f a b -> Star f a b # return :: a -> Star f a a # fail :: String -> Star f a a #
Functor f => Functor (Star f a) # |
Methods fmap :: (a -> b) -> Star f a a -> Star f a b # (<$) :: a -> Star f a b -> Star f a a #
Applicative f => Applicative (Star f a) # |
Methods pure :: a -> Star f a a # (<*>) :: Star f a (a -> b) -> Star f a a -> Star f a b # (*>) :: Star f a a -> Star f a b -> Star f a b # (<*) :: Star f a a -> Star f a b -> Star f a a #
Alternative f => Alternative (Star f a) # |
Methods empty :: Star f a a # (<|>) :: Star f a a -> Star f a a -> Star f a a # some :: Star f a a -> Star f a [a] # many :: Star f a a -> Star f a [a] #
MonadPlus f => MonadPlus (Star f a) # |
Methods mzero :: Star f a a # mplus :: Star f a a -> Star f a a -> Star f a a #
Distributive f => Distributive (Star f a) # |
Methods distribute :: Functor f => f (Star f a a) -> Star f a (f a) # collect :: Functor f => (a -> Star f a b) -> f a -> Star f a (f b) # distributeM :: Monad m => m (Star f a a) -> Star f a (m a) # collectM :: Monad m => (a -> Star f a b) -> m a -> Star f a (m b) #
type Rep (Star f) # |
type Rep (Star f) = f

newtype Costar f d c #

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

Constructors

Costar |
Fields runCostar :: f d -> c

Instances

Functor f => Profunctor (Costar f) # |
Methods dimap :: (a -> b) -> (c -> d) -> Costar f b c -> Costar f a d # lmap :: (a -> b) -> Costar f b c -> Costar f a c # rmap :: (b -> c) -> Costar f a b -> Costar f a c # (#.) :: Coercible * c b => (b -> c) -> Costar f a b -> Costar f a c # (.#) :: Coercible * b a => Costar f b c -> (a -> b) -> Costar f a c #
Functor f => Costrong (Costar f) # |
Methods unfirst :: Costar f (a, d) (b, d) -> Costar f a b # unsecond :: Costar f (d, a) (d, b) -> Costar f a b #
Functor f => Closed (Costar f) # |
Methods closed :: Costar f a b -> Costar f (x -> a) (x -> b) #
Applicative f => Cochoice (Costar f) # |
Methods unleft :: Costar f (Either a d) (Either b d) -> Costar f a b # unright :: Costar f (Either d a) (Either d b) -> Costar f a b #
Traversable w => Choice (Costar w) # |
Methods left' :: Costar w a b -> Costar w (Either a c) (Either b c) # right' :: Costar w a b -> Costar w (Either c a) (Either c b) #
Functor f => Corepresentable (Costar f) # |
Associated Types type Corep (Costar f :: * -> * -> *) :: * -> * # Methods cotabulate :: (Corep (Costar f) d -> c) -> Costar f d c #
Functor f => Cosieve (Costar f) f # |
Methods cosieve :: Costar f a b -> f a -> b #
Monad (Costar f a) # |
Methods (>>=) :: Costar f a a -> (a -> Costar f a b) -> Costar f a b # (>>) :: Costar f a a -> Costar f a b -> Costar f a b # return :: a -> Costar f a a # fail :: String -> Costar f a a #
Functor (Costar f a) # |
Methods fmap :: (a -> b) -> Costar f a a -> Costar f a b # (<$) :: a -> Costar f a b -> Costar f a a #
Applicative (Costar f a) # |
Methods pure :: a -> Costar f a a # (<*>) :: Costar f a (a -> b) -> Costar f a a -> Costar f a b # (*>) :: Costar f a a -> Costar f a b -> Costar f a b # (<*) :: Costar f a a -> Costar f a b -> Costar f a a #
Distributive (Costar f d) # |
Methods distribute :: Functor f => f (Costar f d a) -> Costar f d (f a) # collect :: Functor f => (a -> Costar f d b) -> f a -> Costar f d (f b) # distributeM :: Monad m => m (Costar f d a) -> Costar f d (m a) # collectM :: Monad m => (a -> Costar f d b) -> m a -> Costar f d (m b) #
type Corep (Costar f) # |
type Corep (Costar f) = f

newtype WrappedArrow p a b #

Wrap an arrow for use as a `Profunctor`.

Constructors

WrapArrow |
Fields unwrapArrow :: p a b

Instances

Arrow p => Arrow (WrappedArrow p) # |
Methods arr :: (b -> c) -> WrappedArrow p b c # first :: WrappedArrow p b c -> WrappedArrow p (b, d) (c, d) # second :: WrappedArrow p b c -> WrappedArrow p (d, b) (d, c) # (***) :: WrappedArrow p b c -> WrappedArrow p b' c' -> WrappedArrow p (b, b') (c, c') # (&&&) :: WrappedArrow p b c -> WrappedArrow p b c' -> WrappedArrow p b (c, c') #
ArrowZero p => ArrowZero (WrappedArrow p) # |
Methods zeroArrow :: WrappedArrow p b c #
ArrowChoice p => ArrowChoice (WrappedArrow p) # |
Methods left :: WrappedArrow p b c -> WrappedArrow p (Either b d) (Either c d) # right :: WrappedArrow p b c -> WrappedArrow p (Either d b) (Either d c) # (+++) :: WrappedArrow p b c -> WrappedArrow p b' c' -> WrappedArrow p (Either b b') (Either c c') # (|||) :: WrappedArrow p b d -> WrappedArrow p c d -> WrappedArrow p (Either b c) d #
ArrowApply p => ArrowApply (WrappedArrow p) # |
Methods app :: WrappedArrow p (WrappedArrow p b c, b) c #
ArrowLoop p => ArrowLoop (WrappedArrow p) # |
Methods loop :: WrappedArrow p (b, d) (c, d) -> WrappedArrow p b c #
Arrow p => Profunctor (WrappedArrow p) # |
Methods dimap :: (a -> b) -> (c -> d) -> WrappedArrow p b c -> WrappedArrow p a d # lmap :: (a -> b) -> WrappedArrow p b c -> WrappedArrow p a c # rmap :: (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (#.) :: Coercible * c b => (b -> c) -> WrappedArrow p a b -> WrappedArrow p a c # (.#) :: Coercible * b a => WrappedArrow p b c -> (a -> b) -> WrappedArrow p a c #
ArrowLoop p => Costrong (WrappedArrow p) # |
Methods unfirst :: WrappedArrow p (a, d) (b, d) -> WrappedArrow p a b # unsecond :: WrappedArrow p (d, a) (d, b) -> WrappedArrow p a b #
Arrow p => Strong (WrappedArrow p) # | `Arrow` is `Strong` `Category`
Methods first' :: WrappedArrow p a b -> WrappedArrow p (a, c) (b, c) # second' :: WrappedArrow p a b -> WrappedArrow p (c, a) (c, b) #
ArrowChoice p => Choice (WrappedArrow p) # |
Methods left' :: WrappedArrow p a b -> WrappedArrow p (Either a c) (Either b c) # right' :: WrappedArrow p a b -> WrappedArrow p (Either c a) (Either c b) #
Category * p => Category * (WrappedArrow p) # |
Methods id :: cat a a # (.) :: cat b c -> cat a b -> cat a c #

newtype Forget r a b #

Constructors

Forget |
Fields runForget :: a -> r

Instances

Profunctor (Forget r) # |
Methods dimap :: (a -> b) -> (c -> d) -> Forget r b c -> Forget r a d # lmap :: (a -> b) -> Forget r b c -> Forget r a c # rmap :: (b -> c) -> Forget r a b -> Forget r a c # (#.) :: Coercible * c b => (b -> c) -> Forget r a b -> Forget r a c # (.#) :: Coercible * b a => Forget r b c -> (a -> b) -> Forget r a c #
Strong (Forget r) # |
Methods first' :: Forget r a b -> Forget r (a, c) (b, c) # second' :: Forget r a b -> Forget r (c, a) (c, b) #
Monoid r => Choice (Forget r) # |
Methods left' :: Forget r a b -> Forget r (Either a c) (Either b c) # right' :: Forget r a b -> Forget r (Either c a) (Either c b) #
Representable (Forget r) # |
Associated Types type Rep (Forget r :: * -> * -> *) :: * -> * # Methods tabulate :: (d -> Rep (Forget r) c) -> Forget r d c #
Sieve (Forget r) (Const * r) # |
Methods sieve :: Forget r a b -> a -> Const * r b #
Functor (Forget r a) # |
Methods fmap :: (a -> b) -> Forget r a a -> Forget r a b # (<$) :: a -> Forget r a b -> Forget r a a #
Foldable (Forget r a) # |
Methods fold :: Monoid m => Forget r a m -> m # foldMap :: Monoid m => (a -> m) -> Forget r a a -> m # foldr :: (a -> b -> b) -> b -> Forget r a a -> b # foldr' :: (a -> b -> b) -> b -> Forget r a a -> b # foldl :: (b -> a -> b) -> b -> Forget r a a -> b # foldl' :: (b -> a -> b) -> b -> Forget r a a -> b # foldr1 :: (a -> a -> a) -> Forget r a a -> a # foldl1 :: (a -> a -> a) -> Forget r a a -> a # toList :: Forget r a a -> [a] # null :: Forget r a a -> Bool # length :: Forget r a a -> Int # elem :: Eq a => a -> Forget r a a -> Bool # maximum :: Ord a => Forget r a a -> a # minimum :: Ord a => Forget r a a -> a # sum :: Num a => Forget r a a -> a # product :: Num a => Forget r a a -> a #
Traversable (Forget r a) # |
Methods traverse :: Applicative f => (a -> f b) -> Forget r a a -> f (Forget r a b) # sequenceA :: Applicative f => Forget r a (f a) -> f (Forget r a a) # mapM :: Monad m => (a -> m b) -> Forget r a a -> m (Forget r a b) # sequence :: Monad m => Forget r a (m a) -> m (Forget r a a) #
type Rep (Forget r) # |
type Rep (Forget r) = Const * r

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