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 d a -> q a c -> Procompose p q d c

Instances

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

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

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

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

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

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

Profunctor q => Functor (Procompose p q a)

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

Bicategorical Associators

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.

Generalized Composition

upstars :: Functor f => Iso (Procompose (UpStar f) (UpStar g) d c) (Procompose (UpStar f') (UpStar g') d' c') (UpStar (Compose f g) d c) (UpStar (Compose f' g') 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).

upstars :: Functor f => Iso' (Procompose (UpStar f) (UpStar g) d c) (UpStar (Compose f g) d c)

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

This is a variant on upstars that uses Kleisli instead of UpStar.

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

downstars :: Functor g => Iso (Procompose (DownStar f) (DownStar g) d c) (Procompose (DownStar f') (DownStar g') d' c') (DownStar (Compose g f) d c) (DownStar (Compose g' f') 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.

downstars :: Functor f => Iso' (Procompose (DownStar f) (DownStar g) d c) (DownStar (Compose g f) d c)

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

This is a variant on downstars that uses Cokleisli instead of DownStar.

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