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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 ?

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