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It is well knows that for a nice (locally path connected, semi-locally simply connected) topological spaces, the category of covering spaces over $X$ is equivalent to the functor category $\left[\Pi_1\left(X\right),\mathrm{Set}\right]$ where $\Pi_1\left(X\right)$ is the fundamental groupoid of $X$ (Fundamental Theorem of Covering Spaces).

In general, is the category of covering spaces of a topological space always a

  1. Presheaf category? If yes, is there a natural domain category?
  2. Grothendieck topos? If yes, is there a natural site?
  3. Elementary topos?

Maybe for special sorts of spaces, we get results. I would like to know about those too.

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  • $\begingroup$ So I really don't think the category of covering spaces itself has this kind of properties in general. However, there are modification of it that are much closer to be a topos. I don't remember the technical details enough to give a good answer but I would suggest having a look at this paper matwbn.icm.edu.pl/ksiazki/fm/fm156/fm15611.pdf $\endgroup$
    – Simon Henry
    Commented Jan 6, 2021 at 17:41
  • $\begingroup$ arxiv.org/abs/0706.1771 might also be relevant (if my understanding is correct, it generalizes the result of the paper I cited above from the topos Sh(X) to a general Grothendieck topos) $\endgroup$
    – Simon Henry
    Commented Jan 6, 2021 at 17:41
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    $\begingroup$ The category of covering spaces of $X$ is equivalent to the category of locally constant sheaves on $X$. Maybe things are clearer from that perspective as we then have a direct comparison to a Grothendieck topos (namely : sheaves on $X$). They don't seem to contain the subobject classifier of $Sh(X)$ though, so if one wants to prove that it is an elementary topos one would have to cook up a subobject classifier $\endgroup$
    – Maxime Ramzi
    Commented Jan 6, 2021 at 18:11
  • $\begingroup$ @MaximeRamzi, yes. That is why I thought it may be a Grothendieck topos over a nontrivial site. $\endgroup$
    – Chetan Vuppulury
    Commented Jan 6, 2021 at 18:37

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