Data.Profunctor.Composition

Profunctor Composition

data Procompose p q d c where

Procompose p q is the Profunctor composition of the Profunctors p and q.

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

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

Constructors

Procompose :: p x c -> q d x -> Procompose p q d c

Instances

Category * p => ProfunctorMonad (Procompose p)

Methods

proreturn :: Profunctor p => p :-> Procompose p p

projoin :: Profunctor p => Procompose p (Procompose p p) :-> Procompose p p

ProfunctorFunctor (Procompose p)

Methods

promap :: Profunctor p => (p :-> q) -> Procompose p p :-> Procompose p q

ProfunctorAdjunction (Procompose p) (Rift p)

Methods

unit :: Profunctor p => p :-> Rift p (Procompose p p)

counit :: Profunctor p => Procompose p (Rift p p) :-> p

(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

(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

(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)

(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)

(Closed p, Closed q) => Closed (Procompose p q)

Methods

closed :: Procompose p q a b -> Procompose p q (x -> a) (x -> b)

(Traversing p, Traversing q) => Traversing (Procompose p q)

Methods

traverse' :: Traversable f => Procompose p q a b -> Procompose p q (f a) (f b)

wander :: (forall f. Applicative f => (a -> f b) -> s -> f t) -> Procompose p q a b -> Procompose p q s t

(Mapping p, Mapping q) => Mapping (Procompose p q)

Methods

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

(Corepresentable p, Corepresentable q) => Corepresentable (Procompose p q)

Associated Types

type Corep (Procompose p q :: * -> * -> *) :: * -> *

Methods

cotabulate :: (Corep (Procompose p q) d -> c) -> Procompose p q d c

(Representable p, Representable q) => Representable (Procompose p q)

The composition of two Representable Profunctors is Representable by the composition of their representations.

Associated Types

type Rep (Procompose p q :: * -> * -> *) :: * -> *

Methods

tabulate :: (d -> Rep (Procompose p q) c) -> Procompose p q d c

(Cosieve p f, Cosieve q g) => Cosieve (Procompose p q) (Compose * * f g)

Methods

cosieve :: Procompose p q a b -> Compose * * f g a -> b

(Sieve p f, Sieve q g) => Sieve (Procompose p q) (Compose * * g f)

Methods

sieve :: Procompose p q a b -> a -> Compose * * g f b

Profunctor p => Functor (Procompose p q a)

Methods

fmap :: (a -> b) -> Procompose p q a a -> Procompose p q a b

(<$) :: a -> Procompose p q a b -> Procompose p q a a

type Corep (Procompose p q)

type Corep (Procompose p q) = Compose * * (Corep p) (Corep q)

type Rep (Procompose p q)

type Rep (Procompose p q) = Compose * * (Rep q) (Rep p)

procomposed :: Category p => Procompose p p a b -> p a b

Unitors and Associator

idl :: Profunctor q => Iso (Procompose (->) q d c) (Procompose (->) r d' c') (q d c) (r d' c')

(->) functions as a lax identity for Profunctor composition.

This provides an Iso for the lens package that witnesses the isomorphism between Procompose (->) q d c and q d c, which is the left identity law.

idl :: Profunctor q => Iso' (Procompose (->) q d c) (q d c)

idr :: Profunctor q => Iso (Procompose q (->) d c) (Procompose r (->) d' c') (q d c) (r d' c')

(->) functions as a lax identity for Profunctor composition.

This provides an Iso for the lens package that witnesses the isomorphism between Procompose q (->) d c and q d c, which is the right identity law.

idr :: Profunctor q => Iso' (Procompose q (->) d c) (q d c)

assoc :: Iso (Procompose p (Procompose q r) a b) (Procompose x (Procompose y z) a b) (Procompose (Procompose p q) r a b) (Procompose (Procompose x y) z a b)

The associator for Profunctor composition.

This provides an Iso for the lens package that witnesses the isomorphism between Procompose p (Procompose q r) a b and Procompose (Procompose p q) r a b, which arises because Prof is only a bicategory, rather than a strict 2-category.

Categories as monoid objects

eta :: (Profunctor p, Category p) => (->) :-> p

a Category that is also a Profunctor is a Monoid in Prof

mu :: Category p => Procompose p p :-> p

Generalized Composition

stars :: Functor g => Iso (Procompose (Star f) (Star g) d c) (Procompose (Star f') (Star g') d' c') (Star (Compose g f) d c) (Star (Compose g' f') d' c')

Profunctor composition generalizes Functor composition in two ways.

This is the first, which shows that exists b. (a -> f b, b -> g c) is isomorphic to a -> f (g c).

stars :: Functor f => Iso' (Procompose (Star f) (Star g) d c) (Star (Compose f g) d c)

kleislis :: Monad g => Iso (Procompose (Kleisli f) (Kleisli g) d c) (Procompose (Kleisli f') (Kleisli g') d' c') (Kleisli (Compose g f) d c) (Kleisli (Compose g' f') d' c')

This is a variant on stars that uses Kleisli instead of Star.

kleislis :: Monad f => Iso' (Procompose (Kleisli f) (Kleisli g) d c) (Kleisli (Compose f g) d c)

costars :: Functor f => Iso (Procompose (Costar f) (Costar g) d c) (Procompose (Costar f') (Costar g') d' c') (Costar (Compose f g) d c) (Costar (Compose f' g') d' c')

Profunctor composition generalizes Functor composition in two ways.

This is the second, which shows that exists b. (f a -> b, g b -> c) is isomorphic to g (f a) -> c.

costars :: Functor f => Iso' (Procompose (Costar f) (Costar g) d c) (Costar (Compose g f) d c)

cokleislis :: Functor f => Iso (Procompose (Cokleisli f) (Cokleisli g) d c) (Procompose (Cokleisli f') (Cokleisli g') d' c') (Cokleisli (Compose f g) d c) (Cokleisli (Compose f' g') d' c')

This is a variant on costars that uses Cokleisli instead of Costar.

cokleislis :: Functor f => Iso' (Procompose (Cokleisli f) (Cokleisli g) d c) (Cokleisli (Compose g f) d c)

Right Kan Lift

newtype Rift p q a b

This represents the right Kan lift of a Profunctor q along a Profunctor p in a limited version of the 2-category of Profunctors where the only object is the category Hask, 1-morphisms are profunctors composed and compose with Profunctor composition, and 2-morphisms are just natural transformations.

Constructors

Rift

Fields

• runRift :: forall x. p b x -> q a x

Instances

Category * p => ProfunctorComonad (Rift p)

Methods

proextract :: Profunctor p => Rift p p :-> p

produplicate :: Profunctor p => Rift p p :-> Rift p (Rift p p)

ProfunctorFunctor (Rift p)

Methods

promap :: Profunctor p => (p :-> q) -> Rift p p :-> Rift p q

ProfunctorAdjunction (Procompose p) (Rift p)

Methods

unit :: Profunctor p => p :-> Rift p (Procompose p p)

counit :: Profunctor p => Procompose p (Rift p p) :-> p

(~) (* -> * -> *) p q => Category * (Rift p q)

Rift p p forms a Monad in the Profunctor 2-category, which is isomorphic to a Haskell Category instance.

Methods

id :: cat a a

(.) :: cat b c -> cat a b -> cat 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 => Functor (Rift p q a)

Methods

fmap :: (a -> b) -> Rift p q a a -> Rift p q a b

(<$) :: a -> Rift p q a b -> Rift p q a a

decomposeRift :: Procompose p (Rift p q) :-> q

The 2-morphism that defines a left Kan lift.

Note: When p is right adjoint to Rift p (->) then decomposeRift is the counit of the adjunction.