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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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

(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 …

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 …

As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma …

This has nothing to do with $\infty$-categories, but with the fact that we look at the full topos and not an arbitrary site of definition: Theorem: If $T$ is a (Grothendieck) ($1$-)topos, then a "she …

My question is : Is there a classifying $\infty$-topos for $\infty$-connected objects ? Does this $\infty$-topos has a nice description (as an $\infty$-category ) ? What I mean by $\infty$-connected …

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 …

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 …