Data.Profunctor

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/

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 (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 (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 (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)

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)

Distributive f => Closed (Star f)

Methods

closed :: Star f a b -> Star f (x -> a) (x -> 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

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 => Closed (Costar f)

Methods

closed :: Costar f a b -> Costar f (x -> a) (x -> 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