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](https://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_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?
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> 
> 


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.



Is Mazur's analogy between arithmetic and topology formal, in any sense?
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](https://mathoverflow.net/questions/413011/is-mazurs-analogy-between-arithmetic-and-topology-formal-in-any-sense#comment1058517_413011)), 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.


