TITLE: Are $\infty$-topoi determined by their localic points ?
QUESTION [12 upvotes]: 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 !

REPLY [6 votes]: The functor is conservative if $T$ is hypercomplete. This follows from DAG VII, Cor. 4.14, which says that any $\infty$-topos admits a surjection from a hypercomplete locale (where $f$ is a surjection if $f^\ast$ detects $\infty$-connective morphisms).