Step by Step / Deep explain: The Power of (Co)Yoneda (preferably in scala) through Coroutines
- by Mzk
some background code
/** FunctorStr: ? F[-]. (? A B. (A -> B) -> F[A] -> F[B]) */
trait FunctorStr[F[_]] { self =>
def map[A, B](f: A => B): F[A] => F[B]
}
trait Yoneda[F[_], A] { yo =>
def apply[B](f: A => B): F[B]
def run: F[A] =
yo(x => x)
def map[B](f: A => B): Yoneda[F, B] = new Yoneda[F, B] {
def apply[X](g: B => X) = yo(f andThen g)
}
}
object Yoneda {
implicit def yonedafunctor[F[_]]: FunctorStr[({ type l[x] = Yoneda[F, x] })#l] =
new FunctorStr[({ type l[x] = Yoneda[F, x] })#l] {
def map[A, B](f: A => B): Yoneda[F, A] => Yoneda[F, B] =
_ map f
}
def apply[F[_]: FunctorStr, X](x: F[X]): Yoneda[F, X] = new Yoneda[F, X] {
def apply[Y](f: X => Y) = Functor[F].map(f) apply x
}
}
trait Coyoneda[F[_], A] { co =>
type I
def fi: F[I]
def k: I => A
final def map[B](f: A => B): Coyoneda.Aux[F, B, I] =
Coyoneda(fi)(f compose k)
}
object Coyoneda {
type Aux[F[_], A, B] = Coyoneda[F, A] { type I = B }
def apply[F[_], B, A](x: F[B])(f: B => A): Aux[F, A, B] =
new Coyoneda[F, A] {
type I = B
val fi = x
val k = f
}
implicit def coyonedaFunctor[F[_]]: FunctorStr[({ type l[x] = Coyoneda[F, x] })#l] =
new CoyonedaFunctor[F] {}
trait CoyonedaFunctor[F[_]] extends FunctorStr[({type l[x] = Coyoneda[F, x]})#l] {
override def map[A, B](f: A => B): Coyoneda[F, A] => Coyoneda[F, B] =
x => apply(x.fi)(f compose x.k)
}
def liftCoyoneda[T[_], A](x: T[A]): Coyoneda[T, A] =
apply(x)(a => a)
}
Now I thought I understood yoneda and coyoneda a bit just from the types –
i.e.
that they quantify / abstract over map fixed in some type constructor
F and some type a, to any type B returning F[B] or (Co)Yoneda[F, B].
Thus providing map fusion for free (? is this kind of like a cut rule for map ?).
But I see that Coyoneda is a functor for any type constructor F regardless of F being a Functor,
and that I don't fully grasp.
Now I'm in a situation where I'm trying to define a Coroutine type,
(I'm looking at https://www.fpcomplete.com/school/to-infinity-and-beyond/pick-of-the-week/coroutines-for-streaming/part-2-coroutines for the types to get started with)
case class Coroutine[S[_], M[_], R](resume: M[CoroutineState[S, M, R]])
sealed trait CoroutineState[S[_], M[_], R]
object CoroutineState {
case class Run[S[_], M[_], R](x: S[Coroutine[S, M, R]]) extends CoroutineState[S, M, R]
case class Done[R](x: R) extends CoroutineState[Nothing, Nothing, R]
class CoroutineStateFunctor[S[_], M[_]](F: FunctorStr[S]) extends
FunctorStr[({ type l[x] = CoroutineState[S, M, x]})#l] {
override def map[A, B](f : A => B) : CoroutineState[S, M, A] => CoroutineState[S, M, B]
=
{ ??? }
}
}
and I think that if I understood Coyoneda better I could leverage it to make S & M type constructors functors way easy, plus I see Coyoneda potentially playing a role in defining recursion schemes
as the functor requirement is pervasive.
So how could I use coyoneda to make type constructors functors like for example coroutine state?
or something like a Pause functor ?
