Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a scheme whose small site is homeomorphic to the small site associated to $S^1$ (considered as a topological space)? By a homeomorphism of sites I mean a continuous equivalence of the underlying categories (with a continuous inverse).

See here for some remarks about etale morphisms (I do not think that etale site would work, however).

  • $\begingroup$ Do you mean the subset of closed points? Otherwise the answer is clearly no... $\endgroup$ – abx May 4 '19 at 8:25
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    $\begingroup$ Answers here explain exactly which spaces are underlying spaces of a scheme. $S^n$ isn't such a space, since its quasicompact open subsets don't form a basis. I think this also will work for the space of closed points which abx is mentioning. $\endgroup$ – Wojowu May 4 '19 at 8:52
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    $\begingroup$ The edit doesn't change the fundamental obstruction (the underlying locale of the étale topos of a scheme is a locally spectral space, so it cannot be $S^1$). $\endgroup$ – Denis Nardin May 12 '19 at 18:59
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    $\begingroup$ What is your goal in asking this question? $\endgroup$ – S. Carnahan May 13 '19 at 12:45
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    $\begingroup$ @S.Carnahan just for fun $\endgroup$ – user138661 May 13 '19 at 13:07

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