Natural Transformations 

Ordinary scala functions are monomorphic, which is to say that they transform values of one type to those of another type. The requirements of several problems are however to transform contexts, form example, from a List to an Option, or from an open connection to a closed connection. Natural transformations encode these transformations whilst maintaining the type of the object in the context.

trait ~>[-F[_], +G[_]] {
  def apply[A](a: F[A]): G[A]
}

Since the names of anonymous classes are not particularly interesting and dollar signs confuse the templating engine used to produce the documentation, the names of anonymous classes are replaced by /**/ below.

Supported operations are:

Construction 

scala> import typequux._; // package
import typequux._

scala> import Typequux._ // useful imports
import Typequux._

scala> val singletonList = new (Id ~> List){override def apply[T](t: T) = List(t)}
singletonList: typequux.~>[typequux.typequux.Id,List] = /**/

scala> val list2Option = new (List ~> Option){override def apply[T](x: List[T]) = x.headOption}
list2Option: typequux.~>[List,Option] = /**/

scala> singletonList(12)
res0: List[Int] = List(12)

scala> singletonList("oogachaka")
res1: List[String] = List(oogachaka)

scala> singletonList((true, 42))
res2: List[(Boolean, Int)] = List((true,42))

scala> list2Option(List(1, 2, 3))
res3: Option[Int] = Some(1)

scala> list2Option(List())
res4: Option[Nothing] = None

Composition 

scala> val list2Set = new (List ~> Set) {override def apply[T](x: List[T]) = x.toSet}
list2Set: typequux.~>[List,Set] = /**/

scala> val singletonSet1 = singletonList andThen list2Set
singletonSet1: typequux.~>[typequux.typequux.Id,Set] = /**/

scala> val singletonSet2 = list2Set compose singletonList
singletonSet2: typequux.~>[typequux.typequux.Id,Set] = /**/

scala> singletonSet1(42)
res5: Set[Int] = Set(42)

scala> singletonSet1(true)
res6: Set[Boolean] = Set(true)

scala> singletonSet2("goku")
res7: Set[String] = Set(goku)

Conversion to monomorphic function values 

scala> Vector(List(1, 2,3), List(100, 200, 100)) map list2Option
res9: scala.collection.immutable.Vector[Option[Int]] = Vector(Some(1), Some(100))

scala> Vector(List(1, 2,3), List(100, 200, 100)) map list2Set
res10: scala.collection.immutable.Vector[Set[Int]] = Vector(Set(1, 2, 3), Set(100, 200))