Aux Pattern

auxpattern.scala

import shapeless._

import scalaz._
import Scalaz._

object console extends App {

  trait Foo[A] {
    type B
    def value: B
  }

  object Foo {
    type Aux[A0, B0] = Foo[A0] { type B = B0 }

    implicit def fi = new Foo[Int] {
      type B = String
      val value = "Foo"
    }
    implicit def fs = new Foo[String] {
      type B = Boolean
      val value = false
    }

  }

  def ciao[T, R](t: T)
                (implicit f: Foo.Aux[T, R],
                          m: Monoid[R]): R = f.value
  val res = ciao(2)
  println(s"res: ${res}")
}

auxpattern2.scala

import scalaz._
import Scalaz._

object console extends App {

  trait Apart[F]{
    type T
    type W[X]

    def apply(f: F): W[T]
  }

  object Apart {
    def apply[F](implicit apart: Apart[F]) = apart

    type Aux[FA, F[_], A] = Apart[FA]{ type T = A; type W[X] = F[X]  }

    implicit def mk[F[_], R]: Aux[F[R], F, R] = new Apart[F[R]]{
      type T = R
      type W[X] = F[X]

      def apply(f: F[R]): W[T] = f
    }
  }  

  def mapZero[Thing, F[_], A](thing: Thing)
                             (implicit apart: Apart.Aux[Thing, F, A],
                                           f: Functor[F], 
                                           m: Monoid[A]): F[A] =
    f.map(apart(thing))(_ => m.zero)

  mapZero(Option(3))
}

Great example, thank you!