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Some equivalences of categories are constructed by explicitly giving a pair of functors that are inverses up to isomorphism. For example, the equivalence between CRing^op and affine schemes is given by the pair (Spec, GlobalSections). I'd say these are "natural", since no choices are made.

Another equivalence of categories is between finite dimensional vector spaces and the category consisting of one vector space of each dimension. The functor in one direction is just the inclusion, but the inverse requires making a bunch of choices. I'd say this is "unnatural".

But my definitions of "natural" and "unnatural" aren't precise. I suppose one of the triumphs of category theory has been the ability to make precise the definition of natural in some contexts. So my question is: how can I make this precise?

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You could try to make it precise by asking what equivalences remain if you deprive yourself of the axiom of choice. The point, I think, is that there are two ways to define an equivalence of categories:

  1. A functor F: C --> D is a strong equivalence if there exist a functor G: D --> C and isomorphisms 1 --> GF, FG--> 1.

  2. A functor F: C --> D is a weak equivalence if it is full, faithful and essentially surjective on objects.

To prove that every strong equivalence is weak is straightforward and does not require any kind of choices to be made. But to prove that every weak equivalence is strong, you have to choose for each object D of D an object C of C such that F(C) is isomorphic to D. That requires a strong form of the axiom of choice: you're making as many choices as there are objects of D.

So, if you work in a logical setup where you have no axiom of choice, or only a weak form of it, then some categories that were previously equivalent will no longer be equivalent. For example, most categories will not be equivalent to their skeleton. (You mentioned this in the case of finite-dimensional vector spaces.) Perhaps you could define the natural equivalences to be those that continue to be equivalences in this world without choice.

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  • $\begingroup$ In fact, in the absence of choice, most categories will not even HAVE a skeleton. $\endgroup$
    – Mike Shulman
    Commented Nov 5, 2009 at 5:15
  • $\begingroup$ Suppose I don't want to deny the axiom of choice. Then is there some extra property or structure that the strong equivalence will have that the weak one won't? The example I'm having in mind is the isomorphism from a fin. dim. vector space to it's double dual. These isomorphisms have the nice property of also being natural transforms between two functors. The isomorphism from a vector space to it's single dual probably doesn't have this property, but I'm not interested in proving that. I want a positive test so I can point to a pair of functors and say they give a strong equivalence. $\endgroup$
    – Dinakar Muthiah
    Commented Nov 5, 2009 at 5:32
  • $\begingroup$ How do you make a natural transformation between a covariant and a contravariant functor? $\endgroup$
    – S. Carnahan ♦
    Commented Nov 5, 2009 at 5:52
  • $\begingroup$ You don't. I'm saying maybe there is a covariant functor out there that on objects takes vector spaces to their single dual (probably not, but i don't know how to prove that). The map from a vector space to its double dual doesn't make explicit mention of functors, but here there is an obvious covariant functor that is the double dual on objects. $\endgroup$
    – Dinakar Muthiah
    Commented Nov 5, 2009 at 13:53
  • $\begingroup$ Dinakar, I wonder if you're getting confused between some different concepts here. It's a matter of fact that the double dual functor from Vect to Vect is isomorphic (=naturally isomorphic) to the identity functor. But that has little to do with the distinction between "natural" and "unnatural", or "strong" and "weak", equivalences of categories. $\endgroup$
    – Tom Leinster
    Commented Nov 5, 2009 at 18:14
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On the other hand to Tom's answer, the distinction between strong and weak equivalences disappears if you use anafunctors instead of functors, which is arguably the "right" way to do category theory in the absence of choice. Someone who takes this point of view would argue that the inverse (ana)functor from FDVect to the category of (say) Euclidean spaces actually doesn't involve any unnatural choices, because every time you have to choose something, all of the things you have to choose from are uniquely canonically isomorphic.

It's precisely analogous to how if a category C has binary products, you need to make a bunch of "choices" to define a product functor C × C → C, but in each case the "category of possible choices" is contractible and so you aren't really making any "contentful" choices at all. And actually, depending on what foundational axiom system you use, your example of a "natural" equivalence might actually involve this sort of "contentless" choice as well. The underying space of Spec R is some subset of P(R) (the prime ideals), but in a categorical set theory like ETCS, the power-object P(R) is only characterized up to isomorphism.

Thus, someone who takes this point of view might claim that the distinction you want to draw is actually an illusory one.

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