What does it mean for a value to "have a functor"?

Question

I'm new to this, and I may be missing something important. I've read part one of Category Theory for Programmers, but the most abstract math I did in university was Group Theory so I'm having to read very slowly.

I would ultimately like to understand the theory to ground my use of the techniques, so whenever I feel like I've made some progress I return to the fantasy-land spec to test myself. This time I felt like I knew where to start: I started with Functor.

From the fantasy-land spec:

Functor

1. u.map(a => a) is equivalent to u (identity)
2. u.map(x => f(g(x))) is equivalent to u.map(g).map(f) (composition)

map method

map :: Functor f => f a ~> (a -> b) -> f b 

A value which has a Functor must provide a map method. The map method takes one argument:

u.map(f) 

1. f must be a function,

   i. If f is not a function, the behaviour of map is unspecified.

   ii.f can return any value.

   iii. No parts of f's return value should be checked.

2. map must return a value of the same Functor

I don't understand the phrase "a value which has a functor".

What does it mean for a value to "have a functor"? I don't think "is a functor" or "belongs to a functor" would make any more sense here.

What is the documentation trying to say about u?

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My understanding is that a functor is a mapping between categories (or in this case from a category to itself). We are working in the category of types, in which objects are types and morphisms are families of functions that take a type to another type, or a type to itself.

As far as I can tell, a functor maps a to Functor a. It has some kind of constructor like,

Functor :: a -> F a

and a map function like,

map :: Functor f => (a -> b) -> f a -> f b

The a refers to any type, and the family of functions (a -> b) refers to all morphisms pointing from any particular type a to any other particular type b. So it doesn't even really make sense to me to distinguish between values which do or do not "have a functor" because if at least one functor exists, then it exists "for" every type... so every value can be mapped by a function that belongs to a morphism between two types that has been lifted by a functor. What I mean is: please show me an example of a value which "does not have a functor".

The functor is the mapping; it isn't the type F a, and it isn't values of the type F a. So in the docs, the value u is not a functor, and I don't think it "has a" functor... it is a value of type F a.

Answer 1

As you say, a functor in programming is a mapping between types, but JavaScript has not types, so you have to imagine them. Here, imagine that the function f goes from type a to type b. The value u is the result of some functor F acting on a. The result of u.map(f) is of the type F b, so you have a mapping from F a (that's the type of u) to F b. Here, map is treated as a method of u.

That's as close as you can get to a functor in JavaScript. (It doesn't help that the same letter f is alternatively used for the function and the functor in the documentation.)