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][1], 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 weds these two areas, which should be responsible for the dictionary. Or at least whether there has been work done to suggest such a framework exists.

  [1]: https://en.wikipedia.org/wiki/Artin%E2%80%93Verdier_duality
  [2]: https://mathoverflow.net/questions/4075/questions-about-analogy-between-spec-z-and-3-manifolds