Hello !

If $T$ is an infinity topos, then you can consider the infinity category of geometric morphism from $Sh_{\infty}(\mathcal{L})$ to $T$ for any locale $\mathcal{L}$. This associate to $T$ an infinity stacks over the category of all locale (at least for the etale topology, but also probably for some stronger topology).

My question is : is there anything know about this functor ? is it fully faithful ? or does it has some kind of "conservativity" properties that could allow to give an answer to the question in the tittle ? Or in the contrary is there example of non trivial infinity topos with no (or not enough) morphism from non trivial locale ?

thank you !

truefor $1$-topoi; one can even characterize the stacks that they represent over the site of locales as "etale complete localic stacks" (See: arxiv.org/abs/1011.6070). – David Carchedi Jan 24 '13 at 19:22