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

2012-04-08T20:23:10Z

Hello !

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 !

Edit (13/10/2012) :

I returned to this question a few days ago, and I have some new idea about it :

One might define a localic $\infty$-groupoid simply as a simplicial locale $\mathcal{L}_n$ (I mean a simplicial object of the category of locale). And then associate an $\infty$-topos to it simply by seeing $\mathcal{L}_n$ as a diagram of localic $\infty$-topos and taking its colimite in the $\infty$-category of $\infty$-topos.

Maybe one will need to assume some "kan complexe" hypothesis on $\mathcal{L}_n$ or maybe every simplical locale is going to be equivalent to one which have enough "kan complexe" properties. But I Think we can deduce (or at least try to ^^ ) a simpliciale model category on simplicale locale from this construction.

The important point is that I have thought about a few example, and it's seems to me that (maybe) if we consider two $\infty$-Topos $\mathcal{A}$,$\mathcal{B}$ which both come from simpliciales locales them $Hom(\mathcal{A},\mathcal{B})$ is always small. Which would show that $\infty$-Groupoide form a strict but really interesting sub-category of the category of $\infty$-topos.

unfortunately, I'm still not comfortable enough with infinity category and model category to answer those question at this time...

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

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