# Is Mazur's analogy between arithmetic and topology formal, in any sense?

I preface my question by admitting I know no algebraic geometry nor algebraic number theory. I do know some algebraic topology. I'm a student.

Recently I learned about sheaf cohomology. Then a little bit of etale cohomology, as much as I could stomach having never studied algebraic geometry. Then I came across Artin-Verdier duality, in particular the notion that $$\mathcal{O}_K$$ is 'like a 3-manifold.' This led me to the interesting area of arithmetic topology that wants to understand some larger 'arithmetic $$\leftrightarrow$$ topology dictionary.'

Now I've done some reading to try to grasp the big picture of arithmetic topology. But I'm unclear on one point:

Is the analogy pursued by this arithmetic $$\leftrightarrow$$ topology dictionary formal, in any sense?

So far I've seen it said how this analogy gives a nice way of thinking about number-theoretic things with topology (e.g. prime ideals are like links, and their factors are like the constituent knots.) The words 'inspire' and 'motivate' are used a lot. And there are precise comparisons to be made between the objects on either side (e.g. the algebraic fundamental group of $$\mathrm{Spec} ~\mathbb{Z}$$ is isomorphic to the classical fundamental group of $$S^3$$.)

But I'd like to know whether there is some larger framework that rigorously explains why this analogy exists.

• It's probably not any more formal than the (much more well-studied, and easier to search up) function field analogy. A dream would be some equivalence of categories between appropriate objects, but I don't think anything like that is known. Jan 3, 2022 at 0:53
• Exodromy, as developped by Barwick, Glasman and Haine is a rather systematic way to relate arithmetic and topology; see arXiv:1807.03281. It does not explain Mazur's analogy in full yet, but it certainly goes in this direction. This is a great refinement of methods that have been used successfully to study existence of rational points or $0$-cycles over number fields, in the work of Harpaz, Schlank, Skorobogatov and Wittenberg (around the Hasse principle, Gauss-Manin obstruction...); see e.g. arXiv:1409.0993 or arXiv:1802.09605. Jan 3, 2022 at 1:21

In particular, one can certainly not get an equivalence of categories between some category of 3-manifolds and some category of number fields (as Wojowu suggests in the comments), or any kind of correspondence between one 3-manifold and one number field, that respects the interesting structure like Artin-Verdier duality. (Thus I think the equivalence of fundamental groups between $$S^3$$ and $$\mathbb Z$$ is a red herring.)
Note that in Verdier duality, a pretty fundamental concept is an orientation. Any 3-manifold is either oriented or has a double cover to be oriented. But from the form of Artin-Verdier duality, for a number field to be oriented, it would have to contain the $$n$$th roots of unity for all $$n$$, which is impossible. So the "oriented double cover" in this setting is actually a cover of infinite degree! Covering spaces and dualizing sheaves are some of the concepts we absolutely do want to match up, so I don't think there's any way to wriggle out of this.