为什么Functor是一个更高级的类型

Question

I have following functor definition

trait Functor[F[_]] { 
    def map[A, B](fa: F[A])(f: A => B): F[B]
}
object ListFunctor extends Functor[List] { //
    def map[A, B](f: A => B)(data: List[A]): List[B] = data map f
}

In scala, it is very common that F is a collection type, such as List,Seq,Option, I would ask why Functor has to be higher kinded type, and what does the type parameter F really mean?

Answer 1

why Functor has to be higher kinded type

Functor has to be higher kinded because we want to abstract over a type parameter which itself takes a type parameter (we call this a type constructor , think Functor[List] as a concrete example).

The type of types that Functor deals with are called "first order kinds", their kind is of * -> * . When you looking at concrete instances for Functor , you see we dont provide the inner type parameter. For example, when you define a functor for List as you did in your example, you define it as a Functor[List] . We're not creating a functor for a proper type (ie List[Int] ), but rather any type contained inside the List . This abstraction brings great power, because you can utilize it for any proper List type (of kind *), be it List[String] , List[Int] , etc..

I always like to refer to the image drawn by Adriaan Moore in his paper "Genrics Of A Higher Kind":

What does the type parameter F really mean

F s sole purpose is to define a contract with the implementer of the Functor . By the signature of F we can deduce what kind of type Functor expects. When we see that it has one "placeholder" ( [_] ) we know, by convention, that this means that F should take a single type parameter. If we think about all the types that take a single type parameter, we can see that there are many, for example List , Option , Try , Future , Task , etc.

For a more broad explanation regarding higher kinded types, see What is a higher kinded type in Scala?

Answer 2

I'd answer this a bit differently. The presence of a (higher-kinded) type parameter F[_] in Functor signature means Functor is a typeclass. Typeclasses is a Scala feature that allows one to add features to existing types without modifying them. So Functor[F[_]] here is not a single type, but a blueprint for many other types. Or, in other words, it is not a single type, but a class of types, hence the word "typeclass".

Now to create a concrete functor, eg functor of list, one needs to pass that concrete type as a type parameter to that blueprint, and that concrete type itself has to have one type parameter, by mathematical definition of functor, if you wish. This is what higher-kinded type F[_] in Functor signature means.