Semigroupal 

Functors, Applicative Functors and Monoids:

So far, when we were mapping functions over functors, we usually mapped functions that take only one parameter. But what happens when we map a function like *, which takes two parameters, over a functor?

scala> import cats._, cats.data._, cats.implicits._
import cats._
import cats.data._
import cats.implicits._
scala> val hs = Functor[List].map(List(1, 2, 3, 4)) ({(_: Int) * (_:Int)}.curried)
hs: List[Int => Int] = List(scala.Function2$$Lambda$2615/1775702934@2d3b9a7e, scala.Function2$$Lambda$2615/1775702934@208bfe13, scala.Function2$$Lambda$2615/1775702934@e72df04, scala.Function2$$Lambda$2615/1775702934@5fe836ff)
scala> Functor[List].map(hs) {_(9)}
res6: List[Int] = List(9, 18, 27, 36)

LYAHFGG:

But what if we have a functor value of Just (3 *) and a functor value of Just 5, and we want to take out the function from Just(3 *) and map it over Just 5?

Meet the Applicative typeclass. It lies in the Control.Applicative module and it defines two methods, pure and <*>.

Cats splits this into Semigroupal, Apply, and Applicative. Here’s the contract for Cartesian:

/**
 * [[Semigroupal]] captures the idea of composing independent effectful values.
 * It is of particular interest when taken together with [[Functor]] - where [[Functor]]
 * captures the idea of applying a unary pure function to an effectful value,
 * calling `product` with `map` allows one to apply a function of arbitrary arity to multiple
 * independent effectful values.
 *
 * That same idea is also manifested in the form of [[Apply]], and indeed [[Apply]] extends both
 * [[Semigroupal]] and [[Functor]] to illustrate this.
 */
@typeclass trait Semigroupal[F[_]] {
  def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
}

Semigroupal defines product function, which produces a pair of (A, B) wrapped in effect F[_] out of F[A] and F[B].

Cartesian law 

Cartesian has a single law called associativity:

trait CartesianLaws[F[_]] {
  implicit def F: Cartesian[F]

  def cartesianAssociativity[A, B, C](fa: F[A], fb: F[B], fc: F[C]): (F[(A, (B, C))], F[((A, B), C)]) =
    (F.product(fa, F.product(fb, fc)), F.product(F.product(fa, fb), fc))
}