Data.Functor.Adjunction

adjunctions-4.3: Adjunctions and representable functors

Copyright | 2008-2013 Edward Kmett
License | BSD
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | experimental
Portability | rank 2 types, MPTCs, fundeps
Safe Haskell | Trustworthy
Language | Haskell98

Data.Functor.Adjunction

Description

Synopsis

- class (Functor f, Representable u) => Adjunction f u | f -> u, u -> f where
  - unit :: a -> u (f a)
  - counit :: f (u a) -> a
  - leftAdjunct :: (f a -> b) -> a -> u b
  - rightAdjunct :: (a -> u b) -> f a -> b
- adjuncted :: (Adjunction f u, Profunctor p, Functor g) => p (a -> u b) (g (c -> u d)) -> p (f a -> b) (g (f c -> d))
- tabulateAdjunction :: Adjunction f u => (f () -> b) -> u b
- indexAdjunction :: Adjunction f u => u b -> f a -> b
- zapWithAdjunction :: Adjunction f u => (a -> b -> c) -> u a -> f b -> c
- zipR :: Adjunction f u => (u a, u b) -> u (a, b)
- unzipR :: Functor u => u (a, b) -> (u a, u b)
- unabsurdL :: Adjunction f u => f Void -> Void
- absurdL :: Void -> f Void
- cozipL :: Adjunction f u => f (Either a b) -> Either (f a) (f b)
- uncozipL :: Functor f => Either (f a) (f b) -> f (Either a b)
- extractL :: Adjunction f u => f a -> a
- duplicateL :: Adjunction f u => f a -> f (f a)
- splitL :: Adjunction f u => f a -> (a, f ())
- unsplitL :: Functor f => a -> f () -> f a

# Documentation

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

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

Nothing

Methods

unit :: a -> u (f a)

counit :: f (u a) -> a

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

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

Instances

Adjunction Identity Identity |
Adjunction ((,) e) ((->) e) |
Adjunction f g => Adjunction (IdentityT f) (IdentityT g) |
Adjunction f u => Adjunction (Free f) (Cofree u) |
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') |
(Adjunction f g, Adjunction f' g') => Adjunction (Sum f f') (Product 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))

`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

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

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

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

A right adjoint functor admits an intrinsic notion of zipping

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

Every functor in Haskell permits unzipping

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

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

absurdL :: Void -> f Void

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

And a left adjoint must be inhabited by exactly one element

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

Every functor in Haskell permits `uncozipping`

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

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

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

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