Contravariant Monoidal

The ContravariantMonoidal type class is for Contravariant functors that can define a product function and a unit function.

import cats.Contravariant

trait ContravariantMonoidal[F[_]] extends Contravariant[F] {
  def unit: F[Unit]

  def product[A, B](fa: F[A], fc: F[B]): F[(A, B)]

  def contramap2[A, B, C](fb: F[B], fc: F[C])(f: A => (B, C)): F[A] =
    contramap(product(fb, fc))(f)

Notice that this allows us to define the contramap2 function, much like the map2 function and the pure function on the Applicative typeclass, but in reverse.

Basically, if you have two contexts F[B] and F[C] for types B and C, as well as a way to produce types B and C simultaneously from a type A, then ContravariantMonoidal allows you to obtain a context F[A] for the type A.

Examples of ContravariantMonoidal instances are Eq and Const, but there are also interesting instances for other types.

Predicates Have ContravariantMonoidal

An example application would be the case of predicates. Consider the type,

import cats._

import cats.implicits._

case class Predicate[A](run: A => Boolean)

Then, we can exhibit a ContravariantMonoidal for Predicate by basing it on the Monoid for Boolean via && as,

implicit val contravariantMonoidalPredicate: ContravariantMonoidal[Predicate] =
  new ContravariantMonoidal [Predicate] {
    def unit: Predicate[Unit] = Predicate[Unit](Function.const(true))

    def product[A, B](fa: Predicate[A], fb: Predicate[B]): Predicate[(A, B)] =
      Predicate(x => &&

    def contramap[A, B](fa: Predicate[A])(f: B => A): Predicate[B] =
      Predicate(x =>

We could have also used false and ||, but the “and” version tends to be a little more convenient for this application.

Just like for Contravariant, we can contramap to pull Predicates back along functions.

case class Money(value: Long)
// defined class Money

def isEven: Predicate[Long] = Predicate(_ % 2 == 0)
// isEven: Predicate[Long]

def isEvenMoney: Predicate[Money] = isEven.contramap(_.value)
// isEvenMoney: Predicate[Money]
// res2: Boolean = false

We can also lift functions contravariantly into the context instead of contramapping repeatedly.

def times2Predicate: Predicate[Long] => Predicate[Long] =
  ContravariantMonoidal[Predicate].liftContravariant((x: Long) => 2*x)
// times2Predicate: Predicate[Long] => Predicate[Long]

def liftMoney: Predicate[Long] => Predicate[Money] =
// liftMoney: Predicate[Long] => Predicate[Money]

def trivial = times2Predicate(isEven)
// trivial: Predicate[Long]
// res3: Boolean = true
// res4: Boolean = true

More interestingly, we can combine multiple predicates using a contramapN.

case class Transaction(value: Money, payee: String)
// defined class Transaction

def isEvan: Predicate[String] = Predicate(_ == "Evan")
// isEvan: Predicate[String]

def isGreaterThan50Dollars: Predicate[Money] = liftMoney(Predicate(_ > 50))
// isGreaterThan50Dollars: Predicate[Money]

def isEvenPaymentToEvanOfMoreThan50 =
  (isEvenMoney, isGreaterThan50Dollars, isEvan).contramapN(
    (trans: Transaction) => (trans.value, trans.value, trans.payee))
// isEvenPaymentToEvanOfMoreThan50: Predicate[Transaction], "Evan"))
// res5: Boolean = true