adjunctions-4.2.2: Adjunctions and representable functors

Copyright2008-2013 Edward Kmett
MaintainerEdward Kmett <>
Portabilityrank 2 types, MPTCs, fundeps
Safe HaskellTrustworthy






class (Functor f, Representable u) => Adjunction f u | f -> u, u -> f where Source

An adjunction between Hask and Hask.

Minimal definition: both unit and counit or both leftAdjunct and rightAdjunct, subject to the constraints imposed by the default definitions that the following laws should hold.

unit = leftAdjunct id
counit = rightAdjunct id
leftAdjunct f = fmap f . unit
rightAdjunct f = counit . fmap f

Any implementation is required to ensure that leftAdjunct and rightAdjunct witness an isomorphism from Nat (f a, b) to Nat (a, g b)

rightAdjunct unit = id
leftAdjunct counit = id

Minimal complete definition



unit :: a -> u (f a) Source

counit :: f (u a) -> a Source

leftAdjunct :: (f a -> b) -> a -> u b Source

rightAdjunct :: (a -> u b) -> f a -> b Source


Adjunction Identity Identity 
Adjunction ((,) e) ((->) e) 
Adjunction f g => Adjunction (IdentityT f) (IdentityT g) 
Adjunction f u => Adjunction (Free f) (Cofree u) 
(Adjunction f g, Adjunction f' g') => Adjunction (Coproduct f f') (Product g g') 
Adjunction w m => Adjunction (EnvT e w) (ReaderT e m) 
Adjunction m w => Adjunction (WriterT s m) (TracedT s w) 
(Adjunction f g, Adjunction f' g') => Adjunction (Compose f' f) (Compose g g') 

adjuncted :: (Adjunction f u, Profunctor p, Functor g) => p (a -> u b) (g (c -> u d)) -> p (f a -> b) (g (f c -> d)) Source

leftAdjunct and rightAdjunct form two halves of an isomorphism.

This can be used with the combinators from the lens package.

adjuncted :: Adjunction f u => Iso' (f a -> b) (a -> u b)

tabulateAdjunction :: Adjunction f u => (f () -> b) -> u b Source

Every right adjoint is representable by its left adjoint applied to a unit element

Use this definition and the primitives in Data.Functor.Representable to meet the requirements of the superclasses of Representable.

indexAdjunction :: Adjunction f u => u b -> f a -> b Source

This definition admits a default definition for the index method of 'Index", one of the superclasses of Representable.

zapWithAdjunction :: Adjunction f u => (a -> b -> c) -> u a -> f b -> c Source

zipR :: Adjunction f u => (u a, u b) -> u (a, b) Source

A right adjoint functor admits an intrinsic notion of zipping

unzipR :: Functor u => u (a, b) -> (u a, u b) Source

Every functor in Haskell permits unzipping

unabsurdL :: Adjunction f u => f Void -> Void Source

A left adjoint must be inhabited, or we can derive bottom.

cozipL :: Adjunction f u => f (Either a b) -> Either (f a) (f b) Source

And a left adjoint must be inhabited by exactly one element

uncozipL :: Functor f => Either (f a) (f b) -> f (Either a b) Source

Every functor in Haskell permits uncozipping

extractL :: Adjunction f u => f a -> a Source

duplicateL :: Adjunction f u => f a -> f (f a) Source

splitL :: Adjunction f u => f a -> (a, f ()) Source

unsplitL :: Functor f => a -> f () -> f a Source