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Comment by mc10 | Hacker News Reader

OCaml functors are not the same as Haskell/F# functors/applicatives (which are essentially one and the same in OCaml if you use named arguments).

Instead, they're roughly "functions from modules to modules": https://dev.realworldocaml.org/functors.html. Functors allow you to replicate much of what classes/objects in an OO language offer, but with static instead of dynamic dispatch.

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I am an F# person rather than OCaml and am not familiar with the compiler, but this seems to be overstating the difference: I thought the OCaml functors are “real” mathematical functors that implemented a structured type polymorphism over the “input type” - they look pretty similar to C++ templates superficially but they work similarly to Haskell type classes. They are “functions from modules to modules” but those functions take types as values.

Real experts should let me know if I am mistaken: I don’t think OCaml and Haskell have strictly the same type system, but they are both similarly more expressive than F# because of this specific type -> type polymorphism that in particular lets you do type-checked monads, etc.

Neither Haskell's nor OCaml's functors are the same as the mathematical one I believe, but both are similar in some way that the word makes sense to be used for their respective feature.

In OCaml, functors map between an abstract module and some concrete one. And in Haskell functors are structures that can have a function that maps over their elements. In math a functor is a map between categories. So you can see how all three kinda make sense as functors, but also are different in exactly what they are.

Both Haskell's and OCaml's functors are special cases of the general mathematical one.

How? What are the categories that OCaml's Hashtbl.Make maps between? What are the morphisms in those categories and how are they mapped?

It maps between a category with HashedType implementations as objects and a category with Make implementations as objects. Unfortunately it's not a very interesting functor because to satisfy the functor laws I think the categories have to be discrete. It's basically a function! (And functions are indeed a special case of functors.)

https://caml.inria.fr/pub/docs/manual-ocaml/libref/Hashtbl.M...

Right, in other words it is just a function. I don't think the categories have to be discrete just to satisfy the functor laws; even ignoring that, what would the morphisms be in principle? I don't think there is an obvious choice, at least.

I think in OCaml, "functor" basically just means "like a function but for modules". They might technically be categorical functors, but it seems quite different to the meaning in Haskell where they are plainly categorical functors (modulo Hask technically not being a category).

Agreed, I'm not sure it's a terribly interesting functor.

Here is one way one might implement a categorical Functor (https://www.reddit.com/r/haskell/comments/eoo16m/base_catego...).

A function S -> T maps a S(ource) type to a T(arget) type, like FunctorOf (-S>) (-T>) does between the source category (-S>) and target category (-T>)

    type  Functor :: forall (s :: Type) (t :: Type). (s -> t) -> Constraint
    class (Category (Src f), Category (Tgt f)) => Functor (f :: s -> t) where
      type Src (f :: s -> t) :: Cat s
      type Tgt (f :: s -> t) :: Cat t
      fmap :: Src f a1 a2 -> Tgt f (f a1) (f a2)

    type FunctorOf :: forall (s :: Type) (t :: Type). Cat s -> Cat t -> (s -> t) -> Constraint
    type FunctorOf src tgt f = (Functor f, Src f ~ src, Tgt f ~ tgt)

The usual endofunctor

    type EndofunctorOf :: forall (ob :: Type). Cat ob -> (ob -> ob) -> Constraint
    type EndofunctorOf @ob cat f = FuntorOf @ob @ob cat cat f

we have in Haskell can be defined as FunctorOf @Type @Type (->) (->), or

    type OldFunctor :: (Type -> Type) -> Constraint
    type OldFunctor f = EndofunctorOf @Type (->) f

For simplicity written with a type synonym, but it cannot be partially applied. So I would use constraint synonym encoding (https://gist.github.com/Icelandjack/5afdaa32f41adf3204ef9025...)

    type FunctorOf :: forall (s :: Type) (t :: Type). Cat s -> Cat t -> Constraint

    class    (Functor f, Src f ~ src, Tgt f ~ tgt) => FunctorOf src tgt f
    instance (Functor f, Src f ~ src, Tgt f ~ tgt) => FunctorOf src tgt f

The other definitions can be eta reduced

    type EndofunctorOf cat = FunctorOf cat cat
    type OldFunctor        = EndofunctorOf (->)

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