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Let me break down the statement you are trying to prove into two independent facts. This answer is not really in the spirit of the question since I will make maximal use of sieves, but for such founda …

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 …

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 …

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, and this doesn't require any assumption on $\mathcal C$. This follows from the following basic fact: if $I$ is filtered and $(A_i)_{i\in I}$ is a diagram of categories with colimit $A$, then the …

The 1-categorical version of motivic spaces is locally cartesian closed but not a quasitopos. The idea is that there exists an $\mathbb A^1$-contractible scheme that is covered by $\mathbb A^1$-rigid …

Every Grothendieck quasitopos is presentable and locally cartesian closed. These are categories of separated presheaves on a site. The simplest example of a site whose separated presheaves do not form …

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'm not sure if that's exactly what you mean by "being motivic", but this long exact sequence comes from a triangle in Voevodsky's category of (integral) motives $DM(\mathbb Q)$. More generally if $ …

Here's a rather convoluted proof of this lemma. It suffices to show that $\mathcal A$ is accessible, that is, $\mathcal A\simeq Ind_\kappa(\mathcal A_0)$ for some small category $\mathcal A_0$ (the su …

It doesn't. For instance, the $\infty$-category of spectra is the colimit of the tower $$ \mathcal{S}_* \stackrel\Sigma\to \mathcal{S}_* \stackrel\Sigma\to ... $$ in $Pr^L$, but its colimit in $Cat_ …

Yes. Colimits in $Pr^L$ are the same as limits in $Pr^R$, which are created by the forgetful functor to $Cat_\infty$ (Higher Topos Theory, 5.5.3.18). So the pushout $E$ of your diagram is the pullback …

Here's a summary of the comments above (which does not answer the question on the origin of the term, that I have no idea). A colimit in a category $C$ with pullbacks is van Kampen if the indexing fu …

This seems to be a special case of Prop. 4.1.2.15 in Higher Topos Theory: Let $p: X \to S$ be a coCartesian fibration of simplicial sets. Then $p$ is smooth. Take $X$ and $S$ to be the nerves of …

Categories $J$ such that limits of shape $J$ commute with filtered colimits in sets are called L-finite. There are several known characterization of them: see the nLab page about it. The page refers t …