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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.

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    $\begingroup$ 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. $\endgroup$
    – Wojowu
    Jan 3, 2022 at 0:53
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    $\begingroup$ 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. $\endgroup$ Jan 3, 2022 at 1:21

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For this analogy, like most analogies in mathematics, and indeed like most philosophical principles in mathematics, one can certainly make a part of it formal and rigorous, but I don't think any true formal statement could ever capture all of what we mean by the analogy.

In particular, by well-chosen definitions, one can write down statements of the form "If X is either a 3-manifold or the ring of integers of a number field, then something is true about X", where "something" is expressed the same way in each case. But there's no reason to expect that there is a single statement that implies all true such statements.

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.

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    $\begingroup$ There is perhaps an equivalence of categories going on, but it won't be exactly the pair (3-manifolds, number fields). Likely you'll have to slightly change both categories to find a proper equivalence. At least, that's my suspicion. Or a more negative way to look at this is that categories may be fairly limited in what they can see, and that a subject area having a few key theorems forces a coarse structure on a category. $\endgroup$ Jan 3, 2022 at 3:14
  • $\begingroup$ In particular, is the statement anything more than the assertion that some cohomological dimension is 3? $\endgroup$
    – Kapil
    Jan 3, 2022 at 4:28
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    $\begingroup$ @Kapil Sure, I'm pretty sure not every space with cohomological dimension 3 has a duality for the category of sheaves with dualizing complex a locally constant sheaf of rank one shifted by 3. $\endgroup$
    – Will Sawin
    Jan 3, 2022 at 4:35
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    $\begingroup$ @RyanBudney Certainly I do not have a proof that no category defined via three-manifolds could ever be equivalent to a category defined via number fields! $\endgroup$
    – Will Sawin
    Jan 3, 2022 at 4:41

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