Q:

In Scala, how do I build a factory for nested types, outside of the containing class?

Explained with an example:
class A {
    def f = {
        val b: B = C.factory(this).asInstanceOf[B]
    }
    class B
}

object C {
    def factory(a: A): A#B = new a.B
}

C.factory is a function for creating a new instance of an A#B. Since B is a nested type, I've included an A reference in the function signature. The calling function f has to cast the reference returned from factory, which I'd like to avoid. Without the cast I get:
error: type mismatch;
found   : A#B
required: A.this.B

A:

It depends on what you want. In A, B means this.B, that is a B that is created from the enclosing instance. Indeed your factory returns that, but it does not say so. It just says it returns the A#B (called a type projection), a B of some unspecified A instance. If, for your val b, you don't care by which A instance it was created, then you should say so with val b: A#B (or let the compiler infer it). 
If you do care that it is a B from your A and no other, you might be out of luck. Your factory returns an instance of B created by the a parameter. But your signature does not says so. You would want a.B rather than the less precise type projection A#B. Alas, the language does not allow that. You will get an error illegal dependent method type: when you write a dependent type a.B, a must be a "stable identifier" and a method parameter is not considered one. Maybe this blog post may help

