类别理论中的应用程序中的地图功能有什么作用？

Question

Given the following statement about functional programming - in particular reasoning about category theory - we see the map function inside the Applicative .

trait Applicative[F[_]] extends Functor[F] {
  def map2[A,B,C](fa: F[A], fb: F[B])(f: (A, B) => C): F[C] =
    apply(map(fa)(f.curried))(fb)
...

My question is: What does the map function in the Applicative from Category theory do?

Answer 1

It's defined just a few lines after map2 , using only apply and unit :

def map[A,B](fa: F[A])(f: A => B): F[B] = apply(unit(f))(fa)

You can plug in the definition of map into map2 and thereby obtain a definition of map2 that relies only on apply and unit :

def map2[X, Y, Z](x: F[X], y: F[Y])(f: (X, Y) => Z): F[Z] = 
  apply(apply(unit((x: X) => (y: Y) => f(x, y)))(x))(y)

Therefore, map is not required to define map2 from apply and unit , because it itself can be derived from apply and unit .

It behaves just like map of any other Functor (because every Applicative is automatically a Functor ): given an F[A] and an f: A => B , it produces an F[B] .