profunctors-5.2.1: Profunctors

Data.Profunctor.Composition

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:

Constructors

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

Instances

 # Methodsproreturn :: Profunctor p => p :-> Procompose p p #projoin :: Profunctor p => Procompose p (Procompose p p) :-> Procompose p p # # Methodspromap :: Profunctor p => (p :-> q) -> Procompose p p :-> Procompose p q # # Methodsunit :: 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) # Methodsdimap :: (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) # Methodsunfirst :: 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) # Methodsfirst' :: 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) # Methodsleft' :: 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) # Methodsclosed :: Procompose p q a b -> Procompose p q (x -> a) (x -> b) # (Traversing p, Traversing q) => Traversing (Procompose p q) # Methodstraverse' :: 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) # Methodsmap' :: Functor f => Procompose p q a b -> Procompose p q (f a) (f b) # # Associated Typestype Corep (Procompose p q :: * -> * -> *) :: * -> * # Methodscotabulate :: (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 Typestype Rep (Procompose p q :: * -> * -> *) :: * -> * # Methodstabulate :: (d -> Rep (Procompose p q) c) -> Procompose p q d c # (Cosieve p f, Cosieve q g) => Cosieve (Procompose p q) (Compose * * f g) # Methodscosieve :: Procompose p q a b -> Compose * * f g a -> b # (Sieve p f, Sieve q g) => Sieve (Procompose p q) (Compose * * g f) # Methodssieve :: Procompose p q a b -> a -> Compose * * g f b # Profunctor p => Functor (Procompose p q a) # Methodsfmap :: (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 FieldsrunRift :: forall x. p b x -> q a x Instances  # Methodsproextract :: Profunctor p => Rift p p :-> p #produplicate :: Profunctor p => Rift p p :-> Rift p (Rift p p) # # Methodspromap :: Profunctor p => (p :-> q) -> Rift p p :-> Rift p q # # Methodsunit :: 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. Methodsid :: cat a a #(.) :: cat b c -> cat a b -> cat a c # (Profunctor p, Profunctor q) => Profunctor (Rift p q) # Methodsdimap :: (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) # Methodsfmap :: (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.