Generalize ApplyBuilder to Monoidal.

core/src/main/scala/cats/Applicative.scala

@@ -28,8 +28,6 @@ import simulacrum.typeclass
    */
   def pureEval[A](x: Eval[A]): F[A] = pure(x.value)
 
-  override def map[A, B](fa: F[A])(f: A => B): F[B] = ap(fa)(pure(f))
-
   /**
    * Two sequentially dependent Applicatives can be composed.
    *

core/src/main/scala/cats/Apply.scala

@@ -8,27 +8,27 @@ import simulacrum.typeclass
  * Must obey the laws defined in cats.laws.ApplyLaws.
  */
 @typeclass(excludeParents=List("ApplyArityFunctions"))
-trait Apply[F[_]] extends Functor[F] with ApplyArityFunctions[F] { self =>
+trait Apply[F[_]] extends Functor[F] with Monoidal[F] with ApplyArityFunctions[F] { self =>
 
   /**
    * Given a value and a function in the Apply context, applies the
    * function to the value.
    */
-  def ap[A, B](fa: F[A])(f: F[A => B]): F[B]
+  def ap[A, B](fa: F[A])(ff: F[A => B]): F[B]
 
   /**
    * ap2 is a binary version of ap, defined in terms of ap.
    */
-  def ap2[A, B, Z](fa: F[A], fb: F[B])(f: F[(A, B) => Z]): F[Z] =
-    ap(fb)(ap(fa)(map(f)(f => (a: A) => (b: B) => f(a, b))))
+  def ap2[A, B, Z](fa: F[A], fb: F[B])(ff: F[(A, B) => Z]): F[Z] =
+    map(product(fa, product(fb, ff))) { case (a, (b, f)) => f(a, b) }
 
   /**
    * Applies the pure (binary) function f to the effectful values fa and fb.
    *
    * map2 can be seen as a binary version of [[cats.Functor]]#map.
    */
   def map2[A, B, Z](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] =
-    ap(fb)(map(fa)(a => (b: B) => f(a, b)))
+    map(product(fa, fb)) { case (a, b) => f(a, b) }
 
   /**
    * Two sequentially dependent Applys can be composed.
@@ -45,6 +45,7 @@ trait Apply[F[_]] extends Functor[F] with ApplyArityFunctions[F] { self =>
       def F: Apply[F] = self
       def G: Apply[G] = GG
     }
+
 }
 
 trait CompositeApply[F[_], G[_]]
@@ -54,4 +55,8 @@ trait CompositeApply[F[_], G[_]]
 
   def ap[A, B](fa: F[G[A]])(f: F[G[A => B]]): F[G[B]] =
     F.ap(fa)(F.map(f)(gab => G.ap(_)(gab)))
+
+  def product[A, B](fa: F[G[A]], fb: F[G[B]]): F[G[(A, B)]] =
+    F.map2(fa, fb)(G.product)
+
 }

core/src/main/scala/cats/FlatMap.scala

@@ -29,6 +29,9 @@ import simulacrum.typeclass
   override def ap[A, B](fa: F[A])(ff: F[A => B]): F[B] =
     flatMap(ff)(f => map(fa)(f))
 
+  override def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] =
+    flatMap(fa)(a => map(fb)(b => (a, b)))
+
   /**
    *  Pair `A` with the result of function application.
    */

core/src/main/scala/cats/Functor.scala

@@ -10,7 +10,8 @@ import functor.Contravariant
  *
  * Must obey the laws defined in cats.laws.FunctorLaws.
  */
-@typeclass trait Functor[F[_]] extends functor.Invariant[F] { self =>
+@typeclass
+trait Functor[F[_]] extends functor.Invariant[F] { self =>
   def map[A, B](fa: F[A])(f: A => B): F[B]
 
   def imap[A, B](fa: F[A])(f: A => B)(fi: B => A): F[B] = map(fa)(f)
@@ -60,6 +61,7 @@ import functor.Contravariant
    * Replaces the `A` value in `F[A]` with the supplied value.
    */
   def as[A, B](fa: F[A], b: B): F[B] = map(fa)(_ => b)
+
 }
 
 object Functor {

core/src/main/scala/cats/Monoidal.scala

@@ -0,0 +1,15 @@
+package cats
+
+import simulacrum.typeclass
+
+/**
+ * Monoidal allows us to express uncurried function application within a context,
+ * whatever the context variance is.
+ *
+ * It is worth noting that the couple Monoidal and [[Functor]] is interdefinable with [[Apply]].
+ */
+@typeclass trait Monoidal[F[_]] {
+  def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
+}
+
+object Monoidal extends MonoidalArityFunctions

core/src/main/scala/cats/data/Const.scala

@@ -76,6 +76,12 @@ sealed abstract class ConstInstances0 extends ConstInstances1 {
 
     def ap[A, B](fa: Const[C, A])(f: Const[C, A => B]): Const[C, B] =
       f.retag[B] combine fa.retag[B]
+
+    def map[A, B](fa: Const[C, A])(f: A => B): Const[C, B] =
+      fa.retag[B]
+
+    def product[A, B](fa: Const[C, A], fb: Const[C, B]): Const[C, (A, B)] =
+      fa.retag[(A, B)] combine fb.retag[(A, B)]
   }
 }
 
@@ -89,6 +95,9 @@ sealed abstract class ConstInstances1 {
     def ap[A, B](fa: Const[C, A])(f: Const[C, A => B]): Const[C, B] =
       fa.retag[B] combine f.retag[B]
 
+    def product[A, B](fa: Const[C, A], fb: Const[C, B]): Const[C, (A, B)] =
+      fa.retag[(A, B)] combine fb.retag[(A, B)]
+
     def map[A, B](fa: Const[C, A])(f: A => B): Const[C, B] =
       fa.retag[B]
   }

core/src/main/scala/cats/data/Func.scala

@@ -54,19 +54,21 @@ abstract class FuncInstances1 {
 
 sealed trait FuncFunctor[F[_], C] extends Functor[Lambda[X => Func[F, C, X]]] {
   def F: Functor[F]
-  override def map[A, B](fa: Func[F, C, A])(f: A => B): Func[F, C, B] =
+  def map[A, B](fa: Func[F, C, A])(f: A => B): Func[F, C, B] =
     fa.map(f)(F)
 }
 
 sealed trait FuncApply[F[_], C] extends Apply[Lambda[X => Func[F, C, X]]] with FuncFunctor[F, C] {
   def F: Apply[F]
-  override def ap[A, B](fa: Func[F, C, A])(f: Func[F, C, A => B]): Func[F, C, B] =
+  def ap[A, B](fa: Func[F, C, A])(f: Func[F, C, A => B]): Func[F, C, B] =
     Func.func(c => F.ap(fa.run(c))(f.run(c)))
+  def product[A, B](fa: Func[F, C, A], fb: Func[F, C, B]): Func[F, C, (A, B)] =
+    Func.func(c => F.product(fa.run(c), fb.run(c)))
 }
 
 sealed trait FuncApplicative[F[_], C] extends Applicative[Lambda[X => Func[F, C, X]]] with FuncApply[F, C] {
   def F: Applicative[F]
-  override def pure[A](a: A): Func[F, C, A] =
+  def pure[A](a: A): Func[F, C, A] =
     Func.func(c => F.pure(a))
 }
 
@@ -119,10 +121,12 @@ abstract class AppFuncInstances {
 
 sealed trait AppFuncApplicative[F[_], C] extends Applicative[Lambda[X => AppFunc[F, C, X]]] {
   def F: Applicative[F]
-  override def map[A, B](fa: AppFunc[F, C, A])(f: A => B): AppFunc[F, C, B] =
+  def map[A, B](fa: AppFunc[F, C, A])(f: A => B): AppFunc[F, C, B] =
     fa.map(f)
-  override def ap[A, B](fa: AppFunc[F, C, A])(f: AppFunc[F, C, A => B]): AppFunc[F, C, B] =
+  def ap[A, B](fa: AppFunc[F, C, A])(f: AppFunc[F, C, A => B]): AppFunc[F, C, B] =
     Func.appFunc[F, C, B](c => F.ap(fa.run(c))(f.run(c)))(F)
-  override def pure[A](a: A): AppFunc[F, C, A] =
+  def product[A, B](fa: AppFunc[F, C, A], fb: AppFunc[F, C, B]): AppFunc[F, C, (A, B)] =
+    Func.appFunc[F, C, (A, B)](c => F.product(fa.run(c), fb.run(c)))(F)
+  def pure[A](a: A): AppFunc[F, C, A] =
     Func.appFunc[F, C, A](c => F.pure(a))(F)
 }

core/src/main/scala/cats/data/Kleisli.scala

@@ -147,6 +147,12 @@ sealed abstract class KleisliInstances1 extends KleisliInstances2 {
 
     def ap[B, C](fa: Kleisli[F, A, B])(f: Kleisli[F, A, B => C]): Kleisli[F, A, C] =
       fa(f)
+
+    def map[B, C](fb: Kleisli[F, A, B])(f: B => C): Kleisli[F, A, C] =
+      fb.map(f)
+
+    def product[B, C](fb: Kleisli[F, A, B], fc: Kleisli[F, A, C]): Kleisli[F, A, (B, C)] =
+      Kleisli.function(a => Applicative[F].product(fb.run(a), fc.run(a)))
   }
 }
 
@@ -155,6 +161,9 @@ sealed abstract class KleisliInstances2 extends KleisliInstances3 {
     def ap[B, C](fa: Kleisli[F, A, B])(f: Kleisli[F, A, B => C]): Kleisli[F, A, C] =
       fa(f)
 
+    def product[B, C](fb: Kleisli[F, A, B], fc: Kleisli[F, A, C]): Kleisli[F, A, (B, C)] =
+      Kleisli.function(a => Apply[F].product(fb.run(a), fc.run(a)))
+
     def map[B, C](fa: Kleisli[F, A, B])(f: B => C): Kleisli[F, A, C] =
       fa.map(f)
   }

core/src/main/scala/cats/data/Prod.scala

@@ -71,20 +71,22 @@ sealed abstract class ProdInstance4 {
 sealed trait ProdFunctor[F[_], G[_]] extends Functor[Lambda[X => Prod[F, G, X]]] {
   def F: Functor[F]
   def G: Functor[G]
-  override def map[A, B](fa: Prod[F, G, A])(f: A => B): Prod[F, G, B] = Prod(F.map(fa.first)(f), G.map(fa.second)(f))
+  def map[A, B](fa: Prod[F, G, A])(f: A => B): Prod[F, G, B] = Prod(F.map(fa.first)(f), G.map(fa.second)(f))
 }
 
 sealed trait ProdApply[F[_], G[_]] extends Apply[Lambda[X => Prod[F, G, X]]] with ProdFunctor[F, G] {
   def F: Apply[F]
   def G: Apply[G]
-  override def ap[A, B](fa: Prod[F, G, A])(f: Prod[F, G, A => B]): Prod[F, G, B] =
+  def ap[A, B](fa: Prod[F, G, A])(f: Prod[F, G, A => B]): Prod[F, G, B] =
     Prod(F.ap(fa.first)(f.first), G.ap(fa.second)(f.second))
+  def product[A, B](fa: Prod[F, G, A], fb: Prod[F, G, B]): Prod[F, G, (A, B)] =
+    Prod(F.product(fa.first, fb.first), G.product(fa.second, fb.second))
 }
 
 sealed trait ProdApplicative[F[_], G[_]] extends Applicative[Lambda[X => Prod[F, G, X]]] with ProdApply[F, G] {
   def F: Applicative[F]
   def G: Applicative[G]
-  override def pure[A](a: A): Prod[F, G, A] = Prod(F.pure(a), G.pure(a))
+  def pure[A](a: A): Prod[F, G, A] = Prod(F.pure(a), G.pure(a))
 }
 
 sealed trait ProdSemigroupK[F[_], G[_]] extends SemigroupK[Lambda[X => Prod[F, G, X]]] {

core/src/main/scala/cats/data/Validated.scala

@@ -109,15 +109,25 @@ sealed abstract class Validated[+E, +A] extends Product with Serializable {
     b => that.fold(_ => false, AA.eqv(b, _))
   )
 
-
   /**
    * From Apply:
    * if both the function and this value are Valid, apply the function
    */
   def ap[EE >: E, B](f: Validated[EE, A => B])(implicit EE: Semigroup[EE]): Validated[EE,B] =
     (this, f) match {
       case (Valid(a), Valid(f)) => Valid(f(a))
-      case (Invalid(e1), Invalid(e2)) => Invalid(EE.combine(e2,e1))
+      case (Invalid(e1), Invalid(e2)) => Invalid(EE.combine(e2, e1))
+      case (e@Invalid(_), _) => e
+      case (_, e@Invalid(_)) => e
+    }
+
+  /**
+   * From Product
+   */
+  def product[EE >: E, B](fb: Validated[EE, B])(implicit EE: Semigroup[EE]): Validated[EE, (A, B)] =
+    (this, fb) match {
+      case (Valid(a), Valid(b)) => Valid((a, b))
+      case (Invalid(e1), Invalid(e2)) => Invalid(EE.combine(e1, e2))
       case (e @ Invalid(_), _) => e
       case (_, e @ Invalid(_)) => e
     }
@@ -189,8 +199,11 @@ sealed abstract class ValidatedInstances extends ValidatedInstances1 {
       override def map[A, B](fa: Validated[E,A])(f: A => B): Validated[E, B] =
         fa.map(f)
 
-      override def ap[A,B](fa: Validated[E,A])(f: Validated[E,A=>B]): Validated[E, B] =
+      def ap[A,B](fa: Validated[E,A])(f: Validated[E,A=>B]): Validated[E, B] =
         fa.ap(f)(E)
+
+      def product[A, B](fa: Validated[E, A], fb: Validated[E, B]): Validated[E, (A, B)] =
+        fa.product(fb)(E)
     }
 }
 

core/src/main/scala/cats/functor/Contravariant.scala

@@ -6,7 +6,8 @@ import simulacrum.typeclass
 /**
  * Must obey the laws defined in cats.laws.ContravariantLaws.
  */
-@typeclass trait Contravariant[F[_]] extends Invariant[F] { self =>
+@typeclass
+trait Contravariant[F[_]] extends Invariant[F] { self =>
   def contramap[A, B](fa: F[A])(f: B => A): F[B]
   override def imap[A, B](fa: F[A])(f: A => B)(fi: B => A): F[B] = contramap(fa)(fi)
 

core/src/main/scala/cats/functor/Invariant.scala

@@ -6,7 +6,8 @@ import simulacrum.typeclass
 /**
  * Must obey the laws defined in cats.laws.InvariantLaws.
  */
-@typeclass trait Invariant[F[_]] extends Any with Serializable { self =>
+@typeclass
+trait Invariant[F[_]] extends Serializable { self =>
   def imap[A, B](fa: F[A])(f: A => B)(g: B => A): F[B]
 
   /**

core/src/main/scala/cats/syntax/all.scala

@@ -3,6 +3,7 @@ package syntax
 
 trait AllSyntax
     extends ApplySyntax
+    with MonoidalSyntax
     with BifunctorSyntax
     with CoflatMapSyntax
     with ComonadSyntax

core/src/main/scala/cats/syntax/apply.scala

@@ -2,31 +2,17 @@ package cats
 package syntax
 
 trait ApplySyntax1 {
-  implicit def applySyntaxU[FA](fa: FA)(implicit U: Unapply[Apply, FA]): ApplyOps[U.M, U.A] =
-    new ApplyOps[U.M, U.A] {
+  implicit def applySyntaxU[FA](fa: FA)(implicit U: Unapply[Apply, FA]): Apply.Ops[U.M, U.A] =
+    new Apply.Ops[U.M, U.A] {
       val self = U.subst(fa)
       val typeClassInstance = U.TC
     }
 }
 
 trait ApplySyntax extends ApplySyntax1 {
-  implicit def applySyntax[F[_], A](fa: F[A])(implicit F: Apply[F]): ApplyOps[F, A] =
-    new ApplyOps[F,A] {
+  implicit def applySyntax[F[_], A](fa: F[A])(implicit F: Apply[F]): Apply.Ops[F, A] =
+    new Apply.Ops[F,A] {
       val self = fa
       val typeClassInstance = F
     }
 }
-
-abstract class ApplyOps[F[_], A] extends Apply.Ops[F, A] {
-  def |@|[B](fb: F[B]): ApplyBuilder[F]#ApplyBuilder2[A, B] = new ApplyBuilder[F] |@| self |@| fb
-
-  /**
-   * combine both contexts but only return the right value
-   */
-  def *>[B](fb: F[B]): F[B] = typeClassInstance.map2(self, fb)((a,b) => b)
-
-  /**
-   * combine both contexts but only return the left value
-   */
-  def <*[B](fb: F[B]): F[A] = typeClassInstance.map2(self, fb)((a,b) => a)
-}

core/src/main/scala/cats/syntax/monoidal.scala

@@ -0,0 +1,28 @@
+package cats
+package syntax
+
+trait MonoidalSyntax1 {
+  implicit def monoidalSyntaxU[FA](fa: FA)(implicit U: Unapply[Monoidal, FA]): MonoidalOps[U.M, U.A] =
+    new MonoidalOps[U.M, U.A] {
+      val self = U.subst(fa)
+      val typeClassInstance = U.TC
+    }
+}
+
+trait MonoidalSyntax extends MonoidalSyntax1 {
+  implicit def monoidalSyntax[F[_], A](fa: F[A])(implicit F: Monoidal[F]): MonoidalOps[F, A] =
+    new MonoidalOps[F, A] {
+      val self = fa
+      val typeClassInstance = F
+    }
+}
+
+abstract class MonoidalOps[F[_], A] extends Monoidal.Ops[F, A] {
+  def |@|[B](fb: F[B]): MonoidalBuilder[F]#MonoidalBuilder2[A, B] =
+    new MonoidalBuilder[F] |@| self |@| fb
+
+  def *>[B](fb: F[B])(implicit F: Functor[F]): F[B] = F.map(typeClassInstance.product(self, fb)) { case (a, b) => b }
+
+  def <*[B](fb: F[B])(implicit F: Functor[F]): F[A] = F.map(typeClassInstance.product(self, fb)) { case (a, b) => a }
+
+}

core/src/main/scala/cats/syntax/package.scala

@@ -3,6 +3,7 @@ package cats
 package object syntax {
   object all extends AllSyntax
   object apply extends ApplySyntax
+  object monoidal extends MonoidalSyntax
   object bifunctor extends BifunctorSyntax
   object coflatMap extends CoflatMapSyntax
   object comonad extends ComonadSyntax

docs/src/main/tut/apply.md

@@ -118,7 +118,7 @@ In order to use it, first import `cats.syntax.all._` or `cats.syntax.apply._`.
 Here we see that the following two functions, `f1` and `f2`, are equivalent:
 
 ```tut
-import cats.syntax.apply._
+import cats.syntax.monoidal._
 
 def f1(a: Option[Int], b: Option[Int], c: Option[Int]) =
   (a |@| b |@| c) map { _ * _ * _ }
@@ -133,7 +133,7 @@ f2(Some(1), Some(2), Some(3))
 All instances created by `|@|` have `map`, `ap`, and `tupled` methods of the appropriate arity:
 
 ```tut
-import cats.syntax.apply._
+import cats.syntax.monoidal._
 
 val option2 = Option(1) |@| Option(2)
 val option3 = option2 |@| Option.empty[Int]

free/src/main/scala/cats/free/FreeApplicative.scala

@@ -81,7 +81,9 @@ object FreeApplicative {
 
   implicit final def freeApplicative[S[_]]: Applicative[FA[S, ?]] = {
     new Applicative[FA[S, ?]] {
-      def ap[A, B](fa: FA[S, A])(f: FA[S, A => B]): FA[S, B] = fa.ap(f)
+      def product[A, B](fa: FA[S, A], fb: FA[S, B]): FA[S, (A, B)] = ap(fb)(fa.map(a => b => (a, b)))
+      def map[A, B](fa: FA[S, A])(f: A => B): FA[S, B] = fa.map(f)
+      override def ap[A, B](fa: FA[S, A])(f: FA[S, A => B]): FA[S, B] = fa.ap(f)
       def pure[A](a: A): FA[S, A] = Pure(a)
     }
   }