The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{-}\mathrm{Topos}$.

Is it true that the $\infty$-category of sheaves on a simplicial locale defines a complete embedding $\mathrm{Sh}: \mathrm{Localic}\infty\text{-}\mathrm{Groupoid} \to \infty\text{-}\mathrm{Topos}$ (where the first $\infty$-category, I expect, can be defined as $[\Delta^{op} , \mathrm{Locale}]$ with a suitable model structure).

If so, how is his image characterized? Maybe these are the $\infty$-topoi that can be obtained by topological localizations?

This looks extremely natural, but I could not find where this is discussed in the literature. I found only the following relevant pages:

• https://ncatlab.org/nlab/show/classifying+topos+of+a+localic+groupoid
• https://ncatlab.org/nlab/show/sheaves+on+a+simplicial+topological+space

I also found a very similar thread 11 years ago: $\infty$-topos and localic $\infty$-groupoids?. But it seems that I have a slightly different accent and in any case, what has become known since then?

The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{-}\mathrm{Topos}$.

Is it true that there is the definition of the $\infty$-topos of simplicial sheaves on a localic $\infty$-groupoid giving a full embedding $\mathrm{Sh}: \mathrm{Localic}\infty\text{-}\mathrm{Groupoid} \to \infty\text{-}\mathrm{Topos}$ (where the first $\infty$-category, I expect, can be defined as $[\Delta^{op} , \mathrm{Locale}]$ with a suitable model structure).

If so, how is his image characterized? Maybe these are the $\infty$-topoi that can be obtained by topological localizations?

This looks extremely natural, but I could not find where this is discussed in the literature. I found only the following relevant pages:

• https://ncatlab.org/nlab/show/classifying+topos+of+a+localic+groupoid
• https://ncatlab.org/nlab/show/sheaves+on+a+simplicial+topological+space

I also found a very similar thread 11 years ago: $\infty$-topos and localic $\infty$-groupoids?. But it seems that I have a slightly different accent and in any case, what has become known since then?

The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ is equivalent to the 2-category $1\text{-}\mathrm{Topos}$. Is it true that the $\infty$-category of sheaves on a simplicial locale defines a complete embedding $\mathrm{Sh}: \mathrm{Localic}\infty\text{-}\mathrm{Groupoid} \to \infty\text{-}\mathrm{Topos}$ (where the first $\infty$-category, I expect, can be defined as $[\Delta^{op} , \mathrm{Locale}]$ with a suitable model structure). If so, how is his image characterized? Maybe these are the $\infty$-topoi that can be obtained by topological localizations?

This looks extremely natural, but I could not find where this is discussed in the literature. I found only the following relevant pages:

• https://ncatlab.org/nlab/show/classifying+topos+of+a+localic+groupoid
• https://ncatlab.org/nlab/show/sheaves+on+a+simplicial+topological+space

I also found a very similar thread 11 years ago: $\infty$-topos and localic $\infty$-groupoids?. But it seems that I have a slightly different accent and in any case, what has become known since then?

The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ . The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{-}\mathrm{Topos}$.

Is it true that the $\infty$-category of sheaves on a simplicial locale defines a complete embedding $\mathrm{Sh}: \mathrm{Localic}\infty\text{-}\mathrm{Groupoid} \to \infty\text{-}\mathrm{Topos}$ (where the first $\infty$-category, I expect, can be defined as $[\Delta^{op} , \mathrm{Locale}]$ with a suitable model structure).

If so, how is his image characterized? Maybe these are the $\infty$-topoi that can be obtained by topological localizations?

This looks extremely natural, but I could not find where this is discussed in the literature. I found only the following relevant pages:

• https://ncatlab.org/nlab/show/classifying+topos+of+a+localic+groupoid
• https://ncatlab.org/nlab/show/sheaves+on+a+simplicial+topological+space

I also found a very similar thread 11 years ago: $\infty$-topos and localic $\infty$-groupoids?. But it seems that I have a slightly different accent and in any case, what has become known since then?