Data.Functor.Yoneda

The covariant form of the Yoneda lemma states that f is naturally isomorphic to Yoneda f.

This is described in a rather intuitive fashion by Dan Piponi in

http://blog.sigfpe.com/2006/11/yoneda-lemma.html

Documentation

newtype Yoneda f a

Yoneda f a can be viewed as the partial application of fmap to its second argument.

Constructors

Yoneda

Fields

runYoneda :: forall b. (a -> b) -> f b

Instances

ComonadTrans Yoneda

MonadTrans Yoneda

(Functor f, MonadFree f m) => MonadFree f (Yoneda m)

Alternative f => Alternative (Yoneda f)

Monad m => Monad (Yoneda m)

Functor (Yoneda f)

MonadFix m => MonadFix (Yoneda m)

MonadPlus m => MonadPlus (Yoneda m)

Applicative f => Applicative (Yoneda f)

Foldable f => Foldable (Yoneda f)

Traversable f => Traversable (Yoneda f)

Distributive f => Distributive (Yoneda f)

Representable g => Representable (Yoneda g)

Comonad w => Comonad (Yoneda w)

Plus f => Plus (Yoneda f)

Traversable1 f => Traversable1 (Yoneda f)

Alt f => Alt (Yoneda f)

Apply f => Apply (Yoneda f)

Bind m => Bind (Yoneda m)

Extend w => Extend (Yoneda w)

Foldable1 f => Foldable1 (Yoneda f)

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

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

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

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

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

type Rep (Yoneda g) = Rep g

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

The natural isomorphism between f and Yoneda f given by the Yoneda lemma is witnessed by liftYoneda and lowerYoneda

liftYoneda . lowerYoneda ≡ id
lowerYoneda . liftYoneda ≡ id

lowerYoneda (liftYoneda fa) =         -- definition
lowerYoneda (Yoneda (f -> fmap f a)) -- definition
(f -> fmap f fa) id                  -- beta reduction
fmap id fa                            -- functor law
fa

lift = liftYoneda

lowerYoneda :: Yoneda f a -> f a

maxF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a

minF :: (Functor f, Ord (f a)) => Yoneda f a -> Yoneda f a -> Yoneda f a

maxM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a

minM :: (Monad m, Ord (m a)) => Yoneda m a -> Yoneda m a -> Yoneda m a

as a right Kan extension

yonedaToRan :: Yoneda f a -> Ran Identity f a

Yoneda f can be viewed as the right Kan extension of f along the Identity functor.

yonedaToRan . ranToYoneda ≡ id
ranToYoneda . yonedaToRan ≡ id

ranToYoneda :: Ran Identity f a -> Yoneda f a

as a right Kan lift

yonedaToRift :: Yoneda f a -> Rift Identity f a

Yoneda f can be viewed as the right Kan lift of f along the Identity functor.

yonedaToRift . riftToYoneda ≡ id
riftToYoneda . yonedaToRift ≡ id

riftToYoneda :: Rift Identity f a -> Yoneda f a