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When I first saw a type lambda in Scala code I thought to myself “What is this?! Line noise?”. Each encounter after the first one wasn’t better and my eyes sort of glazed over and I skipped reading them.

But after the initial shock wore off it turned out that type lambas are really easy to understand if not exactly pleasant looking.

In particular the main idea about type lambda (i.e not the syntax) is really easy to grasp, and we’ll take a look at that in this article.

Why type lambdas?

The need for type lambadas arises when dealing with higher kinded types.

Let’s take a functor type as an example

trait Functor[A, F[_]] {
  def map[B](x: F[A])(f: A => B): F[B]

The F[_] generic parameter is a type of kind * -> * in simple terms a type of kind * -> * is a type constructor that takes another type as a parameter and “returns” a type. Examples include Lists, Options etc.

Coming back to our functor we can easily define an instance for the Option type and see if it works:

implicit def OptionIsFunctor[A]: Functor[A, Option] = new Functor[A, Option] {
  override def map[B](x: Option[A])(f: A => B): Option[B] = x map f

implicitly[Functor[Int, Option]].map(Option(5))(_ + 1)

Now what happens if we try to define a functor for Tuple2 where the map would be mapping on the first argument? We might try something like this:

implicit def Tuple2IsFunctor[X, A](x: Tuple2[X, A]): Functor[A, Tuple2] = new Functor[A, Tuple2] {
  override def map[B](f: A => B): Tuple2[X, B] = (x._1, f(x._2))

But the compiler will rightfully complain that: Tuple2 takes two type parameters, expected: one. What does that mean? Well Tuple2 is of kind (*, *) -> * it’s a type constructor with two type parameters that creates a type, trying to plug in a type constructor of kind * -> * and expecting it to work is not sound.

The hack

So what now? Consider this code:

def Tuple2FunctorTest[X, A](x: Tuple2[X, A]) = {
  type Alias[A] = Tuple2[X, A]
  new Functor[A, Alias] {
    override def map[B](x: Alias[A])(f: A => B): Alias[B] = (x._1, f(x._2))

We created a type alias with one parameter that acts like a one parameter type alias and this is the essence of what the type lambda trick does (believe it or not)! Here’s how this looks:

implicit def Tuple2IsFunctor[X, A]: Functor[A, ({ type Alias[A] = Tuple2[X, A]})#Alias] = 
  new Functor[A, ({ type Alias[A] = Tuple2[X, A]})#Alias] {
    override def map[B](x: Tuple2[X, A])(f: A => B): Tuple2[X, B] = (x._1, f(x._2))

implicitly[Functor[Int, ({ type Alias[A] = Tuple2[String, A] })#Alias]].map(("a", 1))(_ + 1)
// or
type StringTuple[A] = (String, A)
implicitly[Functor[Int, StringTuple]].map(("a", 1))(_ + 1)

To break this down a bit - we need to introduce a type alias and type alias can only be declared in a trait, class, method or an object definition.

Unfortunately we cannot define that somewhere “in between” the def and the return type but what we can do is define a structural type in the generic parameter list.

That’s exactly what the { ... } fragment does. The next part is pretty easy - we define the type alias in the anonymous structural type and then we refer to it with the # operator. And that’s it! Most type lambdas aren’t so wordy and use single letter parameter names (in particular scalaz seems to have a convention of using Greek letters for the type alias something like ({ type λ[α] = Tuple2[X, A] })#λ).