Copyright  (C) 20142015 Edward Kmett 

License  BSDstyle (see the file LICENSE) 
Maintainer  Edward Kmett <ekmett@gmail.com> 
Stability  experimental 
Portability  GADTs, TFs, MPTCs, RankN 
Safe Haskell  None 
Language  Haskell2010 
 data Procompose p q d c where
 Procompose :: p x c > q d x > Procompose p q d c
 procomposed :: Category p => Procompose p p a b > p a b
 idl :: Profunctor q => Iso (Procompose (>) q d c) (Procompose (>) r d' c') (q d c) (r d' c')
 idr :: Profunctor q => Iso (Procompose q (>) d c) (Procompose r (>) d' c') (q d c) (r 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)
 eta :: (Profunctor p, Category p) => (>) :> p
 mu :: Category p => Procompose p p :> p
 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')
 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')
 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')
 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')
 newtype Rift p q a b = Rift {
 runRift :: forall x. p b x > q a x
 decomposeRift :: Procompose p (Rift p q) :> q
Profunctor Composition
data Procompose p q d c where Source #
is the Procompose
p qProfunctor
composition of the
Profunctor
s p
and q
.
For a good explanation of Profunctor
composition in Haskell
see Dan Piponi's article:
Procompose :: p x c > q d x > Procompose p q d c 
procomposed :: Category p => Procompose p p a b > p a b Source #
Unitors and Associator
idl :: 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
and Procompose
(>) q d cq 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') Source #
(>)
functions as a lax identity for Profunctor
composition.
This provides an Iso
for the lens
package that witnesses the
isomorphism between
and Procompose
q (>) d cq 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) Source #
The associator for Profunctor
composition.
This provides an Iso
for the lens
package that witnesses the
isomorphism between
and
Procompose
p (Procompose
q r) a b
, which arises because
Procompose
(Procompose
p q) r a bProf
is only a bicategory, rather than a strict 2category.
Categories as monoid objects
eta :: (Profunctor p, Category p) => (>) :> p Source #
a Category
that is also a Profunctor
is a Monoid
in Prof
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') 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)
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 #
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)
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
This represents the right Kan lift of a Profunctor
q
along a Profunctor
p
in a limited version of the 2category of Profunctors where the only object is the category Hask, 1morphisms are profunctors composed and compose with Profunctor composition, and 2morphisms are just natural transformations.
Category * p => ProfunctorComonad (Rift p) Source #  
ProfunctorFunctor (Rift p) Source #  
ProfunctorAdjunction (Procompose p) (Rift p) Source #  
(~) (* > * > *) p q => Category * (Rift p q) Source # 

(Profunctor p, Profunctor q) => Profunctor (Rift p q) Source #  
Profunctor p => Functor (Rift p q a) Source #  
decomposeRift :: Procompose p (Rift p q) :> q Source #
The 2morphism that defines a left Kan lift.
Note: When p
is right adjoint to
then Rift
p (>)decomposeRift
is the counit
of the adjunction.