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This follow from some recent (I heard about this a year ago) results by Anel, Biedermann, Finster and Joyal. Unfortunately their work is not available yet, but You have some slide of Mathieu Anel on …

My previous answer left open the following: Proposition: Let $C$ be a small $\infty$-category with all fiber products, let $\mathcal{T}$ be an $\infty$-topos and let $F : C \rightarrow \mathcal{T}$ b …

Hello! Is there a simple 'topological' condition to detect whenever a morphism of locales $f : X \rightarrow Y$ induces a surjection of infinity-toposes $f : \mathrm{Sh}_{\infty}(X) \rightarrow \mat …

If you are wiling to assume that $C$ has a terminal object $1 \in C$, which I assume is the case as you said all finite products, you can do the following: (As it is not clear if you are interested i …

I'm currently reading The book of Jacob Lurie, 'Higher Topos Theory', and I'm a little confused by the relation between classical topos and $\infty$-topos : to an $\infty$-topos I can attach the ordi …

As you already noticed there is a functor from the category of Spectrum to the category of topos-spectrum, whose image consist simply of étale topos. As the category of toposes has all limits one eas …

Edit: My original answer contained a big mistake, that I can't fix. A long time I had thought ago about monomorphisms of locales, and I wrongly convince myself that everything would generalizes to top …

(For me the category of toposes is the opposite of the category of left exact left adjoint functors and natural transformations, so $Topos^{co}$ in your sense) The functor $U$ is representable by the …

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site. One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start wi …

Here is an argument for the 1-categorical version that essentially bypass the use of internal site and should be much easier to generalize to the $\infty$-categorical case. ( I mean you can still see …

Let $G$ be a locally compact topological group (or more generally a localic group). Is there an infinity topos which classify principal $G$ bundles ? More precisely, is there an $\infty$-topos $BG$ s …

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e. $$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$ And we want to …

I just found an example, so I thought it would be good to post it here, but if anyone knows other examples, or a more general way to construct some I would be interested to see them as well. This exa …

You need $U$ "contractible". In general $\Delta(S)$ is defined exactly as in the $1$-topos case: "take the constant presheaf valued at $S$ and sheafify it" (i.e. applies the left adjoint to the forget …

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 in …