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If $E$ is an ∞-topos and $T: E \to E$ is an accessible left exact comonad, then indeed the ∞-category $E^T$ of $T$-coalgebras is an ∞-topos. Moreover, it is hypercomplete if $E$ is. I will first show: …

ETA The answer is yes in general. Replace 2 below with a reference to HTT, Prop. 7.1.5.8. Since this has been open for a while, let me give a partial answer which hopefully is already interesting: I …

I believe the answer is YES and, more generally, that $\tau_{\leq n}\mathcal{C}\subset\mathcal{C}$ preserves filtered colimits for any $\infty$-topos $\mathcal{C}$. For the $\infty$-topos of $\infty$- …

If $X_0\to X$ is an effective epimorphism and $X_0$ is locally $n$-coherent, then $X$ is also locally $n$-coherent: every $Y$ over $X$ is covered by $Y\times_XX_0$, which is in turn covered by a copro …

I do not know the answer for a general Quillen adjunction, but I will attempt to give a complete answer in the case you're interested in, when the adjunction $(F,G)$ is of the form $(f_!,f^\ast)$ for …

Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is l …

Yes, they are more general. This is in fact already the case with ordinary rings. Let's call a classically-ringed $\infty$-topos which is locally the Zariski $\infty$-topos of an affine scheme an $\in …

Here's an easy way to resolve the circularity. Proposition 7.2.1.13 is only used in the proof of 7.2.1.14 to establish the following statement: (1) If $f\colon V\to X$ is a monomorphism and is surject …

The plus construction has to be iterated, yes. The topological space from this answer provides a simple counterexample. Let $X=\{a,b,c,d\}$ with opens $\{a\},\{b\},\{a,b\},\{a,b,c\},\{a,b,d\}$. Then t …

There is no relation between hypercompleteness and the property that Čech cohomology agrees with genuine cohomology, i.e., there is no implication either way. For example, étale cohomology of nice sch …

The functor is conservative if $T$ is hypercomplete. This follows from DAG VII, Cor. 4.14, which says that any $\infty$-topos admits a surjection from a hypercomplete locale (where $f$ is a surjection …

I think if $X$ is paracompact of covering dimension $\leq n$ then $\mathrm{Shv}(X)$ is also locally of homotopy dimension $\leq n$: First, the $F_\sigma$ open subsets of $X$ form a basis of the topolo …

This is rarely true. For example, the axioms for ∞-topoi imply that, if $T_S\mathbf H$ is an ∞-topos, then the fibers of $p_S$ must have van Kampen pushouts (more generally van Kampen weakly contracti …

I think your idea to reduce the question to small slice topoi works perfectly. I will use it to show that every sheaf on $Man$ (either the continuous or the smooth version) is the limit of its Postnik …