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I was discussing this with a friend with a deep interest in category theory and homological methods and he said he was pursuing applications of said material in number theory.

I found this rather puzzling since by definition, all possible structures on the set of natural numbers (and any structures constructed from them,such as k-dimensional lattices in n-dimensional Euclidean spaces where k is less then or equal to n) are at most countable.

Would diagram chasing and functorial constructions really give any added information that an ordinary set theoretic construction-using ZFC theory and the usual functions and relations as ordered pairs-give?Have there been uses of category theory in number theory which has lead to deep results that otherwise wouldn't be obvious?

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Andrew, almost any functorial construction you can imagine is useful in number theory. Just open up a book on the subject at the graduate level. And look at finite groups: they're finite but still homological concepts are quite useful there (group cohomology?). Lebesgue said once that if you put everything on an equal footing, without choosing among everything that is exact, then you'd hardly think of many useful concepts. You have to treat the natural numbers as something more than just a countable set to get somewhere in understanding them. –  KConrad Jul 1 '10 at 1:16
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Some find it puzzling that the logarithm has any applications in number theory - but, it does. The countability of the primes has not discouraged the zeta function from having deep things to say about them, even though one must go well beyond the countable and indeed beyond the real to grok zeta. –  Gerry Myerson Jul 1 '10 at 1:19
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Andrew, there are important constructions in number theory, algebraic geometry, and elsewhere that are impossible to make sense of without category theory, so it's not a wrecking ball. Keep in mind that number theory is concerned with structures related to the integers, not only the integers themselves (kind of like real analysis is not the study of real numbers). Please ask your friend for some suggested books or articles that can inform you about the themes which interest him, so you can get a better idea of the hard questions that he would like to study. –  KConrad Jul 1 '10 at 2:34
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One confusion that might be lurking here (and this is only a guess, so I might be totally off base) is your opposition between category theory on one hand and ZFC style set theory on the other. The two are not incompatible! While some category theorists promote it as an alternate set of foundations to ZFC set theory, I don't think that's what most people mean when they talk about using it in number theory, alg geometry, etc. Rather, it is an extremely expressive language for phenomena that reside entirely within standard, ZFC mathematics (indeed, often entirely in the finite world!) –  Andy Putman Jul 1 '10 at 2:44
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Andrew, you seem to be hung up on category theory vs. set theory. They're not incompatible, as Andy wrote, and your instructors (undergraduate?) were working implicitly within ZFC since they wanted to teach you something about the rest of math, so you shouldn't blame them. You're new to diagram chasing, and we all were at one point. You'll get used to it; everyone else did. The point is not to treat the integers or other objects in number theory in isolation, but in a category where the universal qualities of interest are revealed. Nobody does category theory only on the natural numbers. –  KConrad Jul 1 '10 at 3:54

2 Answers 2

Etale cohomology, whose definition and study would be inconceivable without the use of categorical methods, underlies the construction of many interesting Galois representations as well as the entire context for Deligne's work on the Riemann Hypothesis for varieties over finite fields and its various generalizations, and work of Taylor et al. on Sato-Tate.

The concept of moduli schemes and the study of their non-trivial properties (and reasons for their existence) would likewise be impossible without the systematic use of categorical reasoning, and these underlie Faltings' work on the Mordell conjecture, the work of Drinfeld/Lafforgue on global Langlands correspondence for function fields, the work of Wiles et al. on modularity of Galois representations, the work of Mazur/Merel on torsion in elliptic curves over number fields, the use of Heegner points by Gross-Zagier...and the role of Galois cohomology in all of these matters (informed through homological reasoning) is utterly pervasive.

There's so much more which could be said, but I think the above is quite sufficient to make it clear that categorical and homological methods are ubiquitous throughout the deepest parts of modern algebraic number theory.

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Allow me to reinterpret your question as the inquiry

  • How can abstract infinitary constructions inform us about the finite?

To my mind, this is the troubling or at least surprising possibility at the heart of your question, rather than a particular concern about category theory as opposed to other abstract constructions. After all, abstract constructions arise in nearly every area of mathematics. One could bring a sharper focus with the inquiry:

  • How can considerations of the uncountable inform us about the countable or even the finite?

There are, of course, many instances of this, and I am sure that the category theorists will be able to provide interesting examples from category theory. Meanwhile, let me mention several examples of this phenomenon from set theory.

  • Perhaps my way of formulating your question above brings it close to the similar MO question Can infinity shorten proofs a lot? asked by Gowers. Several interesting answers are provided to that questin, among them the observation that considerations of countability and uncountability easily establish the existence of transcendental reals.

  • Another interesting example where considerations of infinity impact the finite arise in Goodstein's Theorem, where one provably must adopt an infinitary theory in order to prove a statement that is ultimately about finite numbers.

  • Large cardinal axioms in set theory provide additional instances where abstract infinitary objects have finitary effects. The reason is that increasingly strong large cardinal assumptions have consistency implications at lower levels of the large cardinal hierarchy that are not provable without those strong assumptions. The point now is that a consistency claim is ultimately a finitary statement about numbers. For each large cardinal notion, there is a diophantine equation that has no solution in the integers if and only if that large cardinal concept is consistent. Thus, one should not expect to find a solution to these equations (since this would refute the large cardinal), but one cannot refute the existence of a solution without making very strong infinitary assumptions.

  • Harvey Friedman has been pursuing his Boolean relation theory, which aims in part to produce natural number theoretic statements that can only be proved on the basis of such large cardinal consistencies. The most natural proofs of these statements lie in the large cardinal infinitary combinatorics, thereby exemplifying the phenomenon.

  • Richard Laver's investigation of the free left-distributive algebra, including a decision procedure for normal forms, first arose out of a consideration of large cardinal concepts at the highest levels of the large carinal hierarchy. (The large cardinals have subsequently been omitted by Dehornoy.) Similar large cardinal considerations arise in the finitary Laver tables.

  • The Axiom of Determinacy, the axiom asserting that in every game of perfect information with countably many binary moves, there is a winning strategy for one of the players, turns out to be intimately connected with Woodin cardinals.

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Thank you for pointing to Laver tables. How amazing! –  Mariano Suárez-Alvarez Jul 1 '10 at 2:57
    
Mariano, your welcome! –  Joel David Hamkins Jul 1 '10 at 3:04
    
I would like to mention that the point of many of my examples is not merely that these infinitary considerations have happened to lead to these finitary conclusions, but rather that the infintary assumptions are provably required for those finitary conclusions. Personally, this makes the examples more compelling than examples merely where abstract methods have been pervasive, but for which we have no proof that those methods are logically required. –  Joel David Hamkins Jul 1 '10 at 3:05
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oops, I mean: you're welcome! –  Joel David Hamkins Jul 1 '10 at 3:20
    
Joel, in "... there is a diophantine equation that has a solution in the integers if and only if that large cardinal concept is consistent," you meant to negate one side of that "if and only if". Consistency means non-existence of a solution. That also affects the next sentence in your answer; in particular, if a Diophantine equation has a solution then of course this can be proved without any large cardinals (indeed, in Robinson's Q). Also, I concur in Mariano's thanks for mentioning the topic of Laver tables. It deserves to be more widely known. –  Andreas Blass Jul 1 '10 at 4:08

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