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The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{-}\mathrm{Topos}$.

Is it true that there is the definition of the $\infty$-topos of simplicial sheaves on a localic $\infty$-groupoid giving a full embedding $\mathrm{Sh}: \mathrm{Localic}\infty\text{-}\mathrm{Groupoid} \to \infty\text{-}\mathrm{Topos}$ (where the first $\infty$-category, I expect, can be defined as $[\Delta^{op} , \mathrm{Locale}]$ with a suitable model structure).

If so, how is his image characterized? Maybe these are the $\infty$-topoi that can be obtained by topological localizations?

This looks extremely natural, but I could not find where this is discussed in the literature. I found only the following relevant pages:

I also found a very similar thread 11 years ago: $\infty$-topos and localic $\infty$-groupoids?. But it seems that I have a slightly different accent and in any case, what has become known since then?

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    $\begingroup$ The natural functor is the embedding $[\Delta^\mathrm{op},\mathrm{Top}_0]\hookrightarrow [\Delta^\mathrm{op},\mathrm{Top}_\infty]$ followed by the colimit over $\Delta^\mathrm{op}$. One knows that every $\infty$-topos is in the essential image up to hypercompletion (Prop. A.4.3.1 in SAG). I think it is generally expected, but not known, that this functor fails to be essentially surjective, in contrast to the classical case (or the case of $n$-topoi for finite $n$). $\endgroup$
    – Marc Hoyois
    Commented Nov 18, 2023 at 7:35
  • $\begingroup$ @MarcHoyois Thank you! Is it unknown whether this functor is a full embedding (and for $n < \infty$ is this an equivalence)? $\endgroup$
    – Arshak Aivazian
    Commented Nov 18, 2023 at 9:08
  • $\begingroup$ It is definitely not fully faithful. The generalization of the classical statement would be that it is a localization onto its essential image. I do not know if this question has been considered before. $\endgroup$
    – Marc Hoyois
    Commented Nov 18, 2023 at 11:56
  • $\begingroup$ I meant with a suitable localization, yes (the phrase "with a suitable model structure" in the text implied that). Okay, but for the rest $1 < n < \infty$ is this theme developed? I mean is there a definition of a $\infty$-category of local $n$-groupoids that is equivalent to a $\infty$-category of $n$-topos? $\endgroup$
    – Arshak Aivazian
    Commented Nov 18, 2023 at 18:57

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