Data.Coyoneda

data CoyonedaF f a i

Coyoneda is encoded as an existential type using Data.Exists.

This type constructor encodes the contents of the existential package.

newtype Coyoneda f a

The Coyoneda Functor.

Coyoneda f is a Functor for any type constructor f. In fact, it is the free Functor for f, i.e. any natural transformation nat :: f ~> g, can be factor through liftCoyoneda. The natural transformation from Coyoneda f ~> g is given by lowerCoyoneda <<< hoistCoyoneda nat:

lowerCoyoneda <<< hoistCoyoneda nat <<< liftCoyoneda $ fi
= lowerCoyoneda (hoistCoyoneda nat (Coyoneda $ mkExists $ CoyonedaF id fi))    (by definition of liftCoyoneda)
= lowerCoyoneda (coyoneda id (nat fi))                                         (by definition of hoistCoyoneda)
= unCoyoneda map (coyoneda id (nat fi))                                        (by definition of lowerCoyoneda)
= unCoyoneda map (Coyoneda $ mkExists $ CoyonedaF  id (nat fi))                (by definition of coyoneda)
= map id (nat fi)                                                              (by definition of unCoyoneda)
= nat fi                                                                       (since g is a Functor)

Constructors

• Coyoneda (Exists (CoyonedaF f a))

Instances

• (Functor f, Eq1 f, Eq a) => Eq (Coyoneda f a)
• (Functor f, Eq1 f) => Eq1 (Coyoneda f)
• (Functor f, Ord1 f, Ord a) => Ord (Coyoneda f a)
• (Functor f, Ord1 f) => Ord1 (Coyoneda f)
• Functor (Coyoneda f)
• (Apply f) => Apply (Coyoneda f)
• (Applicative f) => Applicative (Coyoneda f)
• (Bind f) => Bind (Coyoneda f)
• (Monad f) => Monad (Coyoneda f)

  When f is a Monad then it is a functor as well. In this case liftCoyoneda is not only a functor isomorphism but also a monad isomorphism, i.e. the following law holds

  liftCoyoneda fa >>= liftCoyoneda <<< g = liftCoyoneda $ fa >>= g
  

• MonadTrans Coyoneda
• (Extend w) => Extend (Coyoneda w)
• (Comonad w) => Comonad (Coyoneda w)

  As in the monad case: if w is a comonad, then it is a functor, thus liftCoyoneda is an iso of functors, but moreover it is an iso of comonads, i.e. the following law holds:

  g <<= liftCoyoneda w = liftCoyoneda $ g <<< liftCoyoneda <<= w
  

coyoneda :: forall b a f. (a -> b) -> f a -> Coyoneda f b

Construct a value of type Coyoneda f b from a mapping function and a value of type f a.

unCoyoneda :: forall a g f. (forall b. (b -> a) -> f b -> g a) -> Coyoneda f a -> g a

Deconstruct a value of Coyoneda a to retrieve the mapping function and original value.

liftCoyoneda :: forall f. f ~> (Coyoneda f)

Lift a value described by the type constructor f to Coyoneda f.

Note that for any functor f liftCoyoneda has a right inverse lowerCoyoneda:

liftCoyoneda <<< lowerCoyoneda $ (Coyoneda e)
= liftCoyoneda <<< unCoyoneda map $ (Coyoneda e)
= liftCoyonead (runExists (\(CoyonedaF k fi) -> map k fi) e)
= liftCoyonead (Coyoneda e)
= coyoneda id (Coyoneda e)
= Coyoneda e

Moreover if f is a Functor then liftCoyoneda is an isomorphism of functors with inverse lowerCoyoneda: we already showed that lowerCoyoneda <<< hoistCoyoneda id = lowerCoyoneda is its left inverse whenever f is a functor.

lowerCoyoneda :: forall f. Functor f => (Coyoneda f) ~> f

Lower a value of type Coyoneda f a to the Functor f.

hoistCoyoneda :: forall g f. (f ~> g) -> (Coyoneda f) ~> (Coyoneda g)

Use a natural transformation to change the generating type constructor of a Coyoneda.