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September 10, 2012

The Ax-Grothendieck Theorem According to Category Theory

Posted by David Corfield

The online world of mathematics has taken a considerable interest in Mochizuki’s proposed proof of the ABC conjecture. The best exposition, including some by Minhyong Kim, is in answer to this MO question. An occasional visitor here, Minhyong mentioned Mochizuki’s interest in anabelian geometry to us back here.

I’m bringing this up because at Quomodocumque Terry Tao comments in relation to the announcement of the proof:

I have always been fond of the idea that model-theoretic connections between objects (e.g. relating two objects by comparing the sentences that they satisfy) are at least as important in mathematics as the more traditional category-theoretic connections (where morphisms are the fundamental connective tissue between objects) or topological connections (where the objects are gathered into some common topological space or metric space in order to compare them)…

…A good example is the Ax-Grothendieck theorem, in which a result that is easy to prove in the positive characteristic setting can then be transferred to the characteristic zero setting by a model-theoretic connection, even though there is no immediately obvious morphism, functor, or natural transformation from positive characteristic to zero characteristic, nor is there an immediately obvious topological sense in which the zero characteristic setting is a limit of the positive characteristic one. (This is not to say that there the connection between positive characteristic and zero characteristic that is relevant for Ax-Grothendieck can’t be thought of in categorical or topological terms if one really wanted to view it that way – it probably can – but that the model-theoretic way of looking at it seems much more natural, in my opinion.)

Scheme theory, in my mind, does an excellent job of capturing the categorical and topological ways of connecting objects in algebraic or arithmetic geometry, but only engages in the model-theoretic connections in a rather restricted fashion. (An ideal in a commutative ring can be thought of model-theoretically as the set of all identities that can be deduced from a set of generator identities from the laws of commutative algebra (i.e. high school algebra), and so schemes capture the model theory of this algebra well; but there isn’t an obvious mechanism in place in scheme theory to capture the model theory of more sophisticated theories that might also be relevant in arithmetic geometry.)

Tao also has written on schemes at his own blog, and has some exposition on the Ax-Grothendieck theorem.

We’ve already had a chat about the relationship between Category Theory and Model Theory, looking at two different viewpoints, one by David Kazhdan distancing them, and another by Angus MacIntyre bringing them nearer.

So here’s a very specific question bearing on the issue:

Is there a good category theoretic approach to the kind of model-theoretic reasoning used in the Ax-Grothendieck theorem?

Might moving to world of (,1)(\infty, 1)-categories help? I see someone at MO is wondering something similar about the relation between homotopy type theory and model theory.

Posted at September 10, 2012 10:24 AM UTC

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Re: The Ax-Grothendieck Theorem According to Category Theory

Funnily enough, I’ve been thinking about exactly this over the last couple of weeks, quite independently of the Mochizuki developments: it’s for a paper I’m finishing writing. (Somehow, I don’t think it will attract quite the same attention as Mochizuki’s.) I’ve been reading some of Ax’s paper and looking over my model theory notes from a course by Martin Hyland, which are very much in the style of model theory as “algebraic geometry minus fields”.

One important ingredient in Ax’s proof, as well as other parts of model theory, is the ultraproduct construction. (The definition of ultraproduct can be found here, for example.) Ultraproducts can be used to transfer results between fields of different characteristic, in the way that Tao mentions.

For instance, one theorem in Martin’s notes is this: if a first-order statement ϕ\phi is true in all fields of characteristic zero, then ϕ\phi is true in all fields of sufficiently large characteristic.

This is an easy application of the compactness theorem, which itself can be elegantly proved using ultraproducts, but let’s see how to use ultraproducts to prove this result directly.

We’ll prove the contrapositive. Suppose it is not true that ϕ\phi holds in all fields of sufficiently large characteristic. That is, there are fields of arbitrarily large characteristic in which ϕ\phi fails. Write FF for the first-order theory of fields. Then for each n1n \geq 1, the theory

F{¬ϕ}{20,30,,n0} F \cup \{\neg\phi\} \cup \{ 2 \neq 0, 3 \neq 0, \ldots, n \neq 0 \}

has a model, say k nk_n. Choose a nonprincipal ultrafilter UU on the positive integers, and let kk be the ultraproduct Uk n\prod_U k_n.

I claim that kk is a field of characteristic zero in which ϕ\phi fails. Łoš’s theorem tells us that a statement holds in kk iff the set of nns such that it holds in k nk_n is “large”, i.e. belongs to UU. So:

  • All the field axioms (elements of FF) hold in kk, because they hold in all the k nk_ns. Hence kk is a field.
  • Similarly, ¬ϕ\neg\phi holds in kk.
  • A bit less obviously, kk has characteristic zero; that is, N0N \neq 0 in kk for all N1N \geq 1. That’s because, fixing NN, we have N0N \neq 0 in k nk_n for cofinitely many nn, and the ultrafilter UU (being nonprincipal) contains all cofinite sets.

This proves the claim. So it is not true that ϕ\phi holds in all fields of characteristic zero, as required.

So if we want to bring everything model-theoretic into the categorical fold — and that’s a big “if” — then we might make a start by finding a categorical home for ultraproducts. That was the subject of a MathOverflow question of Joel David Hamkins, which didn’t (in my opinion) get a convincing answer. It’s also something that crops up briefly in my new paper. I don’t have a convincing answer either, but I do have an idea for where the answer might lie, and I’ll write about it here soon.

Posted by: Tom Leinster on September 10, 2012 12:21 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

I wrote this on ultraproducts on an old blog. Not sure it helps much. Maybe from the comments Todd could help us.

You’d think model theoretic compactness would be graspable in category theoretic terms.

Has Makkai not written on compactness or ultraproducts? I see I quote Forssell here

Analogously with the Makkai duality, we consider the groupoid of models and isomorphisms of a theory. Analogously with Stone duality, we use topological structure to equip the models and model isomorphisms of a theory with sufficient structure to recover the theory from them. The result is a first order logical duality which, in comparison to Makkai’s, is more geometrical, in that it uses topology and sheaves on spaces and topological groupoids rather than ultraproducts, and that moreover specializes to the traditional Stone duality.

Posted by: David Corfield on September 10, 2012 1:54 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

I saw that blog entry of yours a little while ago, David: it’s linked to from Joel’s MO question (or rather, from a comment by Qiaochu Yuan just below it).

But nothing I’ve read seems conclusive to me. (Nor, I hasten to add, are any of my own thoughts.) Yes, ultraproducts can be described as colimits in a nice way, but that’s not saying very much — lots of things are nicely described as colimits.

Also, a nice categorical understanding of ultraproducts should include a nice categorical understanding of ultrafilters. Now, there are several nice ways to get at ultrafilters categorically. The trouble is, none of them seems to lead particularly naturally to a categorical understanding of ultraproducts.

What we really want is a single, compelling, story, otherwise there’s no advantage to looking at it categorically at all.

The acid test of a good categorical approach to ultraproducts is that it can gracefully handle change-of-characteristic arguments like the one in my previous comment. I don’t think we’re there yet.

Posted by: Tom Leinster on September 10, 2012 3:20 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

In my opinion, the stress on ultraproducts slightly misses the point. In the applications in the spirit of the Ax-Grothendieck theorem, what is actually used is (as Tom mentions) the compactness theorem. The compactness theorem has the advantage that it does not contain the details of the particular structures used, but only talks about the formulas (which is what you are interested in).

As Tom mentions, to prove the compactness theorem, one could use ultraproducts, but once this is done, it is possible to forget about them for most practical purposes. Thus, the ultraproducts are mostly a technical tool (in my point of view).

So to find a categorical analogue of this kind of results, I would say that a more basic question is to understand the analogues of formulas, structures and the interpretation of formulas. Of course, there are various answers to these questions.

Even if one is really interested in ultraproducts, I would expect that the test that one has a good notion is some analogue of the Los theorem. So a basic question is what is the analogue of the theorem in the categorical setting. This again (probably) requires the notions of formulas and their interpretation.

Posted by: Moshe Kamensky on September 12, 2012 6:36 AM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

No time right now, but I did just find this thesis Ultrasheaves by Jonas Eliasson.

Abstract:

This thesis treats ultrasheaves, sheaves on the category of ultrafilters.

In the classical theory of ultrapowers, you start with an ultrafilter (I, U) and, given a structure S, you construct the ultrapower S I/US^I/U. The fundamental result is Los’s theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure S. In this thesis we instead start with the category of ultrafilters (denoted U). On this category we build the topos Sh(U) of sheaves on U (the ultrasheaves), which we think of as generalized ultrapowers.

The theorem for ultrasheaves corresponding to !Lo´s’s theorem is Moerdijk’s theorem, first proved by Moerdijk for the topos Sh(F) of sheaves on filters. In the thesis we prove that !Lo´s’s theorem follows from Moerdijk’s theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk’s topos Sh(F) and prove that Sh(U) is the double negation subtopos of Sh(F).

The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category U. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result.

One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST), a nonstandard set theory, in the ultrasheaves. IST, introduced by Nelson, is an axiomatic approach to nonstandard mathematics.

Posted by: David Corfield on September 12, 2012 8:50 AM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

There’s also an approach via ‘institutions’, such as Institution-independent Ultraproducts, which gives

an institution-independent generalisation of the so-called ‘Fundamental Theorem on Ultraproducts’ for first-order predicate logic

and

an institution-independent generalisation of the compactness via ultraproducts result.

Posted by: David Corfield on September 13, 2012 2:40 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Moshe, would you say your views on ultraproducts are widely shared in the model theory community? I only ask because there seems to have been a lot of variation in the importance attached to ultraproducts in standard model theory texts.

40 years ago, Chang and Keisler’s book Model Theory devoted one chapter out of seven to ultraproducts. 20 years ago, Hodges’s book Model Theory devoted one amusing page to explaining why ultraproducts had, in his opinion, been overemphasized (section 9.5). I can’t resist quoting part of it:

Some people have hoped […] that ultraproducts might make it possible to do model theory without all the complexities of logic. Disillusionment set in when they found themselves faced with the complexities of boolean algebra instead.

But what do model theorists think today?

I should add that I don’t think everything in mathematics, or even everything in logic, should be thought about categorically. But for something like ultraproducts, they’re so tantalizingly close to the realm of category theory that I think it’s very interesting to try to see whether they can be understood categorically: and if not, why not.

By way of comparison, there’s a standard exercise in beginning category theory that asks for a proof that the centre construction (for groups) does not define a functor. Maybe some people’s immediate interpretation of that is “oh, category theory isn’t as cool as it’s made out to be”. But a more mature understanding is that because so many constructions in algebra are functors, it’s worth investigating when one turns out not to be. (And in the case of centres, it turns out to be rather deep.)

I don’t know what work has been done on the categorical understanding of Łoš’s theorem. I would have thought that early categorical logicians would have thought hard about this, but I’m not familiar with that literature. I’d be interested to hear from anyone who does know it, since I seem not to have got round to looking it up myself.

Posted by: Tom Leinster on September 20, 2012 12:50 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

I didn’t know anything about the Ax-Grothendieck theorem until right now, but after reading only Terry Tao’s blog post (by the way, your link to that needs fixing), it seems to me that his comment

…even though there is no immediately obvious morphism, functor, or natural transformation from positive characteristic to zero characteristic…

is a bit of a straw man. Nobody said that all arguments involving category theory have to use only a single morphism, functor, or natural transformation. It looks to me like the first proof that Tao gives on his blog (which is the one that I guess is the model-theoretic one, though he doesn’t mention any model theory in it) uses two morphisms, which are versions of the two ring homomorphisms \mathbb{Z}\to\mathbb{Q} and /n\mathbb{Z}\to\mathbb{Z}/n with some extra elements adjoined. Does the fact that we have to use a span of ring homomorphisms, rather than a single field homomorphism, to get between characteristic zero and positive characteristic, suddenly make the proof not category-theoretic?

But Terry Tao usually knows what he’s talking about, so I must be missing the point. Maybe it would help to see a version of the proof that uses model theory explicitly?

Posted by: Mike Shulman on September 10, 2012 3:32 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

If you’ve got JSTOR access, Mike, you can find Ax’s original proof here. I don’t know to what extent you’d call it model theory. It’s quite a long paper, and the proof of the relevant result (Theorem C) doesn’t appear until the last page. But if you’re really interested, I can scan and send you the proof from Martin Hyland’s lecture notes, which probably makes the model theory more apparent.

I think Martin proves the result itself in just a page or so, but there’s a lot of buildup involving existentially closed structures, model completeness, etc. Also, he just does the case of endomorphisms of affine space, not an arbitrary algebraic variety.

Posted by: Tom Leinster on September 10, 2012 3:43 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Could you ask Martin to what extent he sees this material as category-theoretic? I wonder how Grothendieck himself did it.

By the way, just seen something of possible interest,

M. Makkai. Ultraproducts and categorical logic. Methods in Mathematical Logic, pages 222-309, 1985,

but we don’t subscribe.

Posted by: David Corfield on September 10, 2012 3:47 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Methods in Mathematical Logic appears to be Springer LNM 1130, which should be in your library somewhere.

Posted by: Finn Lawler on September 10, 2012 5:04 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Well, I don’t really have time to delve into this deeply at the moment. But I’m interested in hearing your thoughts. Are you saying that the special case Tao discusses in his blog post is not typical, and the more general version of the theorem uses more model theory and less category theory?

Posted by: Mike Shulman on September 10, 2012 4:54 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

No, that’s not what I’m saying — though anyway, you shouldn’t attach much weight to what I say, because I have only the sketchiest idea of this stuff. Hopefully, someone who actually knows what they’re talking about will come along.

Grothendieck and Ax proved this theorem independently and by different methods. Ax’s paper is called “The elementary theory of finite fields”, and certainly has lots of logic in it. I’m not enough of a logician to say how much of it should be called model theory. I don’t think Grothendieck used much (any?) logic in his proof.

As far as I know, the amount of model theory involved doesn’t depend significantly on whether we’re looking at the general theorem or the better-known special case that both Terry and Martin discuss. (By the “general theorem” I mean that any injective endomorphism of an algebraic variety over an algebraically closed field is surjective, and by the “special case” I mean the case where the variety is n\mathbb{C}^n.)

No proof of Ax–Grothendieck that I’ve seen involves any nontrivial category theory.

But you should take everything I say about this with a huge pinch of salt.

Posted by: Tom Leinster on September 10, 2012 8:35 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

The more I think about this, the more it seems that this issue can be concentrated by reflecting on ultraproducts. This includes your original point about spans, Mike.

Here’s a basic thing one can do to construct a field of characteristic zero out of fields of positive characteristic. Suppose we have fields k 1,k 2,k 3,k_1, k_2, k_3, \ldots of respective characteristics 2,3,5,2, 3, 5, \ldots. Choose a nonprincipal ultrafilter UU on the positive integers. For each positive integer NN, we have N0N \neq 0 in k nk_n for almost all nn. It follows that the ultraproduct Uk n\prod_U k_n is a field of characteristic zero.

Now, there are no projection maps from an ultraproduct to the original objects. However, there are spans. In this case, whenever jJUj \in J \in U, we have the span of canonical surjections

k j iJk i Uk n. k_j \longleftarrow \prod_{i \in J} k_i \longrightarrow \textstyle{\prod_U} k_n.

In particular, we have this when JJ is the whole set of positive integers.

I don’t know whether this family of spans has some universal property. I guess it does, but I don’t have time to think about it now.

Posted by: Tom Leinster on September 11, 2012 9:08 AM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Is the multispan with projections to all k jk_j as well as the ultraproduct of interest?

Actually, the ultraproduct for a principal ultrafilter is just one of the k jk_j, isn’t it? So what about the multispan from k i\prod k_i to the ultraproducts for each ultrafilter, both principal and nonprincipal?

If the map from a set to its set of ultrafilters has a nice characterisation, maybe that multispan can be universally characterised.

Posted by: David Corfield on September 11, 2012 12:38 PM | Permalink | Reply to this

Re: The Ax-Grothendieck Theorem According to Category Theory

Then there’s all that model theoretic work going on concerning transfer in the proof of the Fundamental Lemma in the Langlands Program. E.g., the work of Cluckers. I wonder how category theoretic that is.

Posted by: David Corfield on September 12, 2012 1:33 PM | Permalink | Reply to this

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