Explore “herding cats”: Functor

5 minute read

Functor extends Invariant.

trait Functor[F[_]] extends Invariant[F]

That’s a little bit tricky once invariant functor is typically explained as mix of usual (covariant) and contravariant functors. Those functors are something like:

trait CovariantFunctor[A]:
  def map[B](f: A => B): CovariantFunctor[B]


trait ContravariantFunctor[A]:
  def contramap[B](f: B => A): ContravariantFunctor[B]

With HKT precise definitions are going to be:

trait CovariantFunctor[F[_]]:
  def map[A, B](fa: F[A])(f: A => B): F[B]


trait ContravariantFunctor[F[_]]:
  def contramap[A, B](fa: F[A])(f: B => A): F[B]

Let’s return back to the notion of Invariant. Cats Invariant doc and Softwaremill Invarian note provide nice examples. Nevertheless, I want to attempt to bring something extremely down-to-earth.

Let we have some thin wrapper:

trait EqWrapper[T]:
  def eqv(valueToCompare: T): Boolean
  def get: T

String implementation:

class StringEqWrapper(private val value: String) extends EqWrapper[String] {
  def eqv(valueToCompare: String): Boolean = valueToCompare == value
  def get: String = value
}

We can assume that in real world we define some non-trivial behaviour. It would be nice to have opportunity to derive new instances from old one using old ones as “back end”.

Let’s attempt to do it with usual map:

class StringEqWrapper(private val value: String) extends EqWrapper[String] { self =>
  def eqv(valueToCompare: String): Boolean = valueToCompare == value
  def get: String = value
  def map[T](f: String => T): EqWrapper[T] = new EqWrapper[T] {
    override def eqv(valueToCompare: T): Boolean = self.eqv("dummy") // I need T => String here to convert value to familiar strings
    override def get: T = f(self.get) //map of basic covariant functor works well
  }
}

At that point we see that String => T helped to implement get. We just apply function to underlying string value and return result.

But we can’t compare T with String.

Contravariant approach leads to opposite result.

class StringEqWrapper2(private val value: String) extends EqWrapper[String] { self =>
  def eqv(valueToCompare: String): Boolean = valueToCompare == value
  def get: String = value
  def map[T](f: T => String): EqWrapper[T] = new EqWrapper[T] {
    override def eqv(valueToCompare: T): Boolean = self.eqv(f(valueToCompare)) // I know how to convert that T value to well-known String
    override def get: T = null.asInstanceOf[T] //I'm in trouble, I have String state but no idea how to return T value
  }
}

We can derive implement eqv(valueToCompare: T) to compare T with internal String state.

But get require something to convert internal String state to T.

class StringEqWrapper(private val value: String) extends EqWrapper[String] { self =>
  def eqv(valueToCompare: String): Boolean = valueToCompare == value
  def get: String = value
  def imap[T](f: String => T, g: T => String): EqWrapper[T] = new EqWrapper[T] {
    override def eqv(value: T): Boolean = self.eqv(g(value))
    override def get: T = f(self.get)
  }
}

Now we can derive new instance with imap:

val stringEqWrapper = new StringEqWrapper3("42")
val intEqWrapper: EqWrapper[Int] = stringEqWrapper.imap(_.toInt, _.toString)

intEqWrapper eqv 42 //true

Typelevel Functor docs provides good description of API.

Explicitly I can denote that it is a right time to pay attention to type lambdas, there is also nice rockthejvm post about it.

import cats.Functor
val listFuntor: Functor[List] = Functor[List]
listFuntor.as(List(1, 2, 3), "a") //List(a, a, a)

val listOfOptionFunctor: Functor[[α] =>> List[Option[α]]] = listFuntor.compose[Option] //Functor[λ[α => F[G[α]]]] in Scala2
listOfOptionFunctor.map(List(Some(1), None))("N" + _) //val res0: List[Option[String]] = List(Some(N1), None)

Functor examples

Anton Kovalevsky