$\infty$-topos and localic $\infty$-groupoids?

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 localic $\infty$-groupoids (I'm not sure of how to define it in a way to be able to construct an $\infty$-topos from it...)

Thank you !

• 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. – David Carchedi Apr 13 '12 at 1:06
• If I found any things about this, I will. But for now I have a lot of thing to learn before... – Simon Henry Apr 13 '12 at 10:43
• So, you're asking if every classical $\infty$-topos is equivalent to the category of sheaves on a localic $\infty$-topos, correct? – user62675 Jun 5 '14 at 0:09
• On a localic $\infty$-groupoid yes. – Simon Henry Jun 5 '14 at 5:51