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?

finitebut 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. $\endgroup$in isolation, but in a category where the universal qualities of interest are revealed. Nobody does category theory only on the natural numbers. $\endgroup$3more comments