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It's known that every classical (Grothendieck) topos is equivalent to the topos of sheaves on a localic groupoid (a groupoid in the category of locales).

For the record, this is proved by, starting form a topos $T$, constructing a locale $L$ and a surjection $L \rightarrow T$ 'nice enough' (like a proper surjection, or an open surjection depending on the proof). Then $(L, L \times_T L, L \times_T L \times_T L)$ is a truncated simplicial locale, which can be seen as a localic groupoid. There is a canonical geometric morphism from the topos of sheaves on this groupoid to $T$, and if the surjection $L \rightarrow T$ was nice enough it's an isomorphism.

My question is : Can we hope for a similar result for $\infty$-toposes ? for example by replacing localic groupoids by a localic $\infty$-groupoids, defined as a simplicial locale satisfying some form of Kan complex condition and defining the associated $\infty$-topos as the colimits of the corresponding simplicial diagram of localic $\infty$-toposes.

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    $\begingroup$ This is something I have thought about in the back of my head for several years. If you come up with something, or would just like to brainstorm, let me know. $\endgroup$ Apr 13, 2012 at 1:06
  • $\begingroup$ If I found any things about this, I will. But for now I have a lot of thing to learn before... $\endgroup$ Apr 13, 2012 at 10:43
  • $\begingroup$ So, you're asking if every classical $\infty$-topos is equivalent to the category of sheaves on a localic $\infty$-topos, correct? $\endgroup$
    – user62675
    Jun 5, 2014 at 0:09
  • $\begingroup$ Do we have a hyperconnected-localic fact system? $\endgroup$ Sep 27, 2022 at 13:45
  • $\begingroup$ @IvanDiLiberti Probably - but as far as I remember it hasn't been written (from what I remember Lurie in HTT talks about n-localic toposes but does not study the relative version) $\endgroup$ Sep 27, 2022 at 14:09

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