Moduleprofunctors-5.6.3Haskell2010

Data.Profunctor.Composition

Contents

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

Synopsis

data Procompose
value procomposed
value idl
value idr
value assoc
value eta
value mu
value stars
value kleislis
value costars
value cokleislis
newtype Rift
value decomposeRift

Packageprofunctors-5.6.3
Exports13
LanguageHaskell2010
LicenceBSD-3-Clause
SourceComposition.hs

Profunctor Composition

datadata Procompose (p :: k -> k1 -> Type) (q :: k2 -> k -> Type) (d :: k2) (c :: k1) where

Source#

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

Procompose has a polymorphic kind since 5.6.

Constructors

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

Instances17ProfunctorFunctor, ProfunctorMonad, ProfunctorAdjunction, Choice, Closed, Mapping, …

ProfunctorFunctor (Procompose p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
Category p => ProfunctorMonad (Procompose p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
ProfunctorAdjunction (Procompose p) (Rift p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Choice p, Choice q) => Choice (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Closed p, Closed q) => Closed (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Mapping p, Mapping q) => Mapping (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Corepresentable p, Corepresentable q) => Costrong (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Strong p, Strong q) => Strong (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Profunctor p, Profunctor q) => Profunctor (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Traversing p, Traversing q) => Traversing (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Corepresentable p, Corepresentable q) => Corepresentable (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Representable p, Representable q) => Representable (Procompose p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
The composition of two Representable Profunctors is Representable by the composition of their representations.
(Cosieve p f, Cosieve q g) => Cosieve (Procompose p q) (Compose f g)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Sieve p f, Sieve q g) => Sieve (Procompose p q) (Compose g f)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
Profunctor p => Functor (Procompose p q a)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
type Corep (Procompose p q) = Compose (Corep p) (Corep q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
type Rep (Procompose p q) = Compose (Rep q) (Rep p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition

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

Unitors and Associator

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

Source#

(->) 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)

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

Source#

(->) 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)

valueassoc

:: (Profunctor p2, Functor f)
=> p2 (Procompose (Procompose p1 q) r a b) (f (Procompose (Procompose x y) z a b))
-> p2 (Procompose p1 (Procompose q r) a b) (f (Procompose x (Procompose y z) a b))

Source#

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

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

Source#

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

valuemu :: Category p => Procompose p p :-> p

Generalized Composition

value

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')

Source#

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)

value

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')

Source#

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)

value

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')

Source#

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)

value

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')

Source#

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

newtypenewtype Rift (p :: k -> k1 -> Type) (q :: k2 -> k1 -> Type) (a :: k2) (b :: k)

Source#

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.

Rift has a polymorphic kind since 5.6.

Constructors

Rift
- runRift :: forall (x :: k2). p b x -> q a x

Instances6ProfunctorFunctor, Category, ProfunctorComonad, ProfunctorAdjunction, Profunctor, Functor

ProfunctorFunctor (Rift p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
p ~ q => Category (Rift p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
Rift p p forms a Monad in the Profunctor 2-category, which is isomorphic to a Haskell Category instance.
Category p => ProfunctorComonad (Rift p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
ProfunctorAdjunction (Procompose p) (Rift p)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
(Profunctor p, Profunctor q) => Profunctor (Rift p q)Defined in profunctors-5.6.3 · Data.Profunctor.Composition
Profunctor p => Functor (Rift p q a)Defined in profunctors-5.6.3 · Data.Profunctor.Composition

valuedecomposeRift :: Procompose p (Rift p q) a b -> q a b

Source#

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.