Data.Functor.Yoneda.Reduction

Yoneda Reduction:

http://ncatlab.org/nlab/show/Yoneda+reduction

Yoneda f is isomorphic to Lan f Identity

Documentation

data Yoneda f a where

A form suitable for Yoneda reduction

Constructors

Yoneda :: (b -> a) -> f b -> Yoneda f a

Instances

ComonadTrans Yoneda

MonadTrans Yoneda

Monad m => Monad (Yoneda m)

Functor (Yoneda f)

MonadFix f => MonadFix (Yoneda f)

MonadPlus f => MonadPlus (Yoneda f)

Applicative f => Applicative (Yoneda f)

Foldable f => Foldable (Yoneda f)

Traversable f => Traversable (Yoneda f)

Alternative f => Alternative (Yoneda f)

Comonad w => Comonad (Yoneda w)

Distributive f => Distributive (Yoneda f)

Keyed f => Keyed (Yoneda f)

Zip f => Zip (Yoneda f)

ZipWithKey f => ZipWithKey (Yoneda f)

(Functor f, Indexable f) => Indexable (Yoneda f)

(Functor f, Lookup f) => Lookup (Yoneda f)

FoldableWithKey f => FoldableWithKey (Yoneda f)

FoldableWithKey1 f => FoldableWithKey1 (Yoneda f)

TraversableWithKey f => TraversableWithKey (Yoneda f)

TraversableWithKey1 f => TraversableWithKey1 (Yoneda f)

Representable f => Representable (Yoneda f)

Plus f => Plus (Yoneda f)

Alt f => Alt (Yoneda f)

Traversable1 f => Traversable1 (Yoneda f)

Foldable1 f => Foldable1 (Yoneda f)

Apply f => Apply (Yoneda f)

Bind m => Bind (Yoneda m)

Extend w => Extend (Yoneda w)

Adjunction f g => Adjunction (Yoneda f) (Yoneda g)

(Functor f, Eq (f a)) => Eq (Yoneda f a)

(Functor f, Ord (f a)) => Ord (Yoneda f a)

(Functor f, Read (f a)) => Read (Yoneda f a)

(Functor f, Show (f a)) => Show (Yoneda f a)

liftYoneda :: f a -> Yoneda f a

Yoneda expansion

 liftYoneda . lowerYoneda ≡ id
 lowerYoneda . liftYoneda ≡ id

 lift = liftYoneda

lowerYoneda :: Functor f => Yoneda f a -> f a

Yoneda reduction

 lower = lowerM = lowerYoneda

lowerM :: Monad f => Yoneda f a -> f a

Yoneda reduction given a Monad.

 lower = lowerM = lowerYoneda

as a Left Kan extension

yonedaToLan :: Yoneda f a -> Lan Identity f a

Yoneda f is the left Kan extension of f along the Identity functor.

 yonedaToLan . lanToYoneda ≡ id
 lanToYoneda . yonedaToLan ≡ id

lanToYoneda :: Lan Identity f a -> Yoneda f a

as a Left Kan lift

yonedaToLift :: Yoneda f a -> Lift Identity f a

Yoneda f is the left Kan lift of f along the Identity functor.

 yonedaToLift . liftToYoneda ≡ id
 liftToYoneda . yonedaToLift ≡ id

liftToYoneda :: Functor f => Lift Identity f a -> Yoneda f a