Data.Functor.Coyoneda

The co-Yoneda lemma for a covariant Functor f states that Coyoneda f is naturally isomorphic to f.

Documentation

data Coyoneda f a where

A covariant Functor suitable for Yoneda reduction

Constructors

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

Instances

ComonadTrans Coyoneda

MonadTrans Coyoneda

Monad m => Monad (Coyoneda m)

Functor (Coyoneda f)

MonadFix f => MonadFix (Coyoneda f)

Applicative f => Applicative (Coyoneda f)

Foldable f => Foldable (Coyoneda f)

Traversable f => Traversable (Coyoneda f)

Distributive f => Distributive (Coyoneda f)

Representable f => Representable (Coyoneda f)

Alternative f => Alternative (Coyoneda f)

MonadPlus f => MonadPlus (Coyoneda f)

Comonad w => Comonad (Coyoneda w)

Plus f => Plus (Coyoneda f)

Traversable1 f => Traversable1 (Coyoneda f)

Alt f => Alt (Coyoneda f)

Apply f => Apply (Coyoneda f)

Bind m => Bind (Coyoneda m)

Extend w => Extend (Coyoneda w)

Foldable1 f => Foldable1 (Coyoneda f)

Adjunction f g => Adjunction (Coyoneda f) (Coyoneda g)

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

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

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

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

type Rep (Coyoneda f) = Rep f

liftCoyoneda :: f a -> Coyoneda f a

Yoneda "expansion"

liftCoyoneda . lowerCoyoneda ≡ id
lowerCoyoneda . liftCoyoneda ≡ id

lowerCoyoneda (liftCoyoneda fa) = -- by definition
lowerCoyoneda (Coyoneda id fa)  = -- by definition
fmap id fa                      = -- functor law
fa

lift = liftCoyoneda

lowerCoyoneda :: Functor f => Coyoneda f a -> f a

Yoneda reduction lets us walk under the existential and apply fmap.

Mnemonically, "Yoneda reduction" sounds like and works a bit like β-reduction.

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

You can view Coyoneda as just the arguments to fmap tupled up.

lower = lowerM = lowerCoyoneda

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

Yoneda reduction given a Monad lets us walk under the existential and apply liftM.

You can view Coyoneda as just the arguments to liftM tupled up.

lower = lowerM = lowerCoyoneda

as a Left Kan extension

coyonedaToLan :: Coyoneda f a -> Lan Identity f a

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

Coyoneda f is always a functor, even if f is not. In this case, it is called the free functor over f. Note the following categorical fine print: If f is not a functor, Coyoneda f is actually not the left Kan extension of f along the Identity functor, but along the inclusion functor from the discrete subcategory of Hask which contains only identity functions as morphisms to the full category Hask. (This is because f, not being a proper functor, can only be interpreted as a categorical functor by restricting the source category to only contain identities.)

coyonedaToLan . lanToCoyoneda ≡ id
lanToCoyoneda . coyonedaToLan ≡ id

lanToCoyoneda :: Lan Identity f a -> Coyoneda f a