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 !

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 a localic $\infty$-groupoids, defined as a simplicial locale satisfying some form of Kan complex condition and defining the associated $\infty$-topos as the colimits of the corresponding simplicial diagram of localic $\infty$-toposes.

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...

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 !

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). 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...)

I'm thinking about it since a few days 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 idea you might spare me a lot of time !

Thank you !

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...