Are there applications of category theory to countable sets?

Question

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?

Answer 1

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.

Answer 2

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 questoin, 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 cardinal hierarchy. The large cardinals have subsequently been omitted from some proofs by Dehornoy, but remain in some results, as of 2011 the proof of "$A_\infty$ is a free group" has not had its large cardinal assumptions removed. (p.66 of Friedman's Boolean Relation Theory, 2011 draft) 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.