In Functional Programming, what is a functor?

Question

I\'ve come across the term \'Functor\' a few times while reading various articles on functional programming, but the authors typically assume the reader already understands

Answer 1

In a comment to the top-voted answer, user Wei Hu asks:

I understand both ML-functors and Haskell-functors, but lack the insight to relate them together. What's the relationship between these two, in a category-theoretical sense?

Note: I don't know ML, so please forgive and correct any related mistakes.

Let's initially assume that we are all familiar with the definitions of 'category' and 'functor'.

A compact answer would be that "Haskell-functors" are (endo-)functors F : Hask -> Hask while "ML-functors" are functors G : ML -> ML'.

Here, Hask is the category formed by Haskell types and functions between them, and similarly ML and ML' are categories defined by ML structures.

Note: There are some technical issues with making Hask a category, but there are ways around them.

From a category theoretic perspective, this means that a Hask-functor is a map F of Haskell types:

data F a = ...

along with a map fmap of Haskell functions:

instance Functor F where
    fmap f = ...

ML is pretty much the same, though there is no canonical fmap abstraction I am aware of, so let's define one:

signature FUNCTOR = sig
  type 'a f
  val fmap: 'a -> 'b -> 'a f -> 'b f
end

That is f maps ML-types and fmap maps ML-functions, so

functor StructB (StructA : SigA) :> FUNCTOR =
struct
  fmap g = ...
  ...
end

is a functor F: StructA -> StructB.