Copyright	(C) 2014-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	GADTs, TFs, MPTCs, RankN
Safe Haskell	None
Language	Haskell2010

Data.Profunctor.Composition

Contents

Profunctor Composition
Unitors and Associator
Categories as monoid objects
Generalized Composition
Right Kan Lift

Description

Synopsis

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)
ProfunctorFunctor (Procompose p)
ProfunctorAdjunction (Procompose p) (Rift p)
(Profunctor p, Profunctor q) => Profunctor (Procompose p q)
(Corepresentable p, Corepresentable q) => Costrong (Procompose p q)
(Strong p, Strong q) => Strong (Procompose p q)
(Closed p, Closed q) => Closed (Procompose p q)
(Choice p, Choice q) => Choice (Procompose p q)
(Corepresentable p, Corepresentable q) => Corepresentable (Procompose p q)
(Representable p, Representable q) => Representable (Procompose p q)	The composition of two `Representable` `Profunctor`s is `Representable` by the composition of their representations.
(Cosieve p f, Cosieve q g) => Cosieve (Procompose p q) (Compose f g)
(Sieve p f, Sieve q g) => Sieve (Procompose p q) (Compose g f)
Profunctor p => Functor (Procompose p q a)
type Corep (Procompose p q) = Compose (Corep p) (Corep 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

(~) (* -> * -> ) 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.
Category * p => ProfunctorComonad (Rift p)
ProfunctorFunctor (Rift p)
ProfunctorAdjunction (Procompose p) (Rift p)
(Profunctor p, Profunctor q) => Profunctor (Rift p q)
Profunctor p => Functor (Rift p q 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.