Apply, Applicative, Monad, etc for contravariant functors in cats?

Question

I have the following trait...

import cats._
import cats.implicits._

trait Preference[-A] {
  self =>

  def compare(a1: A, a2: A): Int

  final def ordering[A1 <: A]: Ordering[A1] = {
    new Ordering[A1] {
      def compare(a1: A1, a2: A1): Int = {
        self.compare(a1, a2)
      }
    }
  }

}


object Preference {

  implicit val contravariant: Contravariant[Preference] = {
    new Contravariant[Preference] {
      def contramap[A, B](fa: Preference[A])(f: B => A): Preference[B] = {
        new Preference[B] {
          def compare(b1: B, b2: B): Int = {
            fa.compare(f(b1), f(b2))
          }
        }
      }
    }
  }
}

I would like to define Apply , Applicative , possibly even Monad instances for this trait but all of these type classes are extensions of Functor . Do versions of these type classes exist in Cats for contravariant functors?

Answer 1

The contravariant equivalent of Applicative in Haskell is Divisible , and cats.ContravariantMonoidal allows to define both of its methods ( divide is contramap2 , conquer is trivial ) . I am not immediately certain, though, if any Divisible has to be Monoidal in cats' sense.

For Monad , Kmett says

There is no contravariant Monad-like. You need C -> C, not C -> C^op. The twist denies you the ability to build nice structure.

Also Are there contravariant monads? .