Hello !

It's know that every classical (Grothendieck) topos is equivalent to the topos of sheaf on a localic groupoid (a groupoid in the category of local).

For the record, this is proved by, starting form a topos $T$, constructing a local $L$ and a surjection $L \rightarrow T$ 'nice enough' (like a proper surjection). then $(L, L \times_T L, L \times_T L \times_T L)$ is a truncatured simplicial local, who can be see as a localic groupoid. There is canonical geometric morphism from the the topos of sheaf on this groupoid to $T$, and if the surjection $L \rightarrow T$ was nice enough it's an isomorphisme.

My question is : Can we hope for a similar result for $\infty$-topos ? for exemple by replacing localic groupoid by localic $\infty$-groupoid (I'm not sure of how to define it in a way to be able to construct an $\infty$-topos from it...)

I'm thinking about it since a few day ago, and I was thinking about using 'localic $\infty$-stacks' but I'm not really familiar with this formalism. So if you know a better way or if you have reason to think that it's not a good a idea you might spare me a lot of time !

Thank you !