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

## 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 $\phi$ is true in all fields of characteristic zero, then $\phi$ 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

nottrue that $\phi$ holds in all fields of sufficiently large characteristic. That is, there are fields of arbitrarily large characteristic in which $\phi$ fails. Write $F$ for the first-order theory of fields. Then for each $n \geq 1$, the theory$F \cup \{\neg\phi\} \cup \{ 2 \neq 0, 3 \neq 0, \ldots, n \neq 0 \}$

has a model, say $k_n$. Choose a nonprincipal ultrafilter $U$ on the positive integers, and let $k$ be the ultraproduct $\prod_U k_n$.

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

allthe $k_n$s. Hence $k$ is a field.This proves the claim. So it is

nottrue that $\phi$ 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.