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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 !

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I don't know the answer on the top of my head. I do know however that this result is true for $1$-topoi; one can even characterize the stacks that they represent over the site of locales as "etale complete localic stacks" (See: – David Carchedi Jan 24 '13 at 19:22
Out of where are you based Simon? – David Carchedi Jan 24 '13 at 19:25
Thank you for your paper, even if i already knew that this was true for classical topos it give really interesting precision on the nature of this embeddings. – Simon Henry Jan 25 '13 at 10:16
My motivation for this question is related to the relation between the geometric objects considered in topos theory/stacks theory and those considered in Non commutative geometry. In the first case (with possibly the exception of infinity topos depending of the answer to this question) every object is completely characterize by its morphism from 'commutative' object. but this is not true in non commutative geometry, for example there exist C^* alegebra of type one, which are not characterize by the stacks of their irreducible representation over all locale. (I'm working in Paris). – Simon Henry Jan 25 '13 at 10:23

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).

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