I wanted to have a better understanding of the geometric interpretation of $\infty$-toposes, and in particular learn something about étale morphisms, but I got stuck trying to unravel two points in the proof of HTT 6.3.5.13.
- At the end of the proof, it is claimed that the $\infty$-category of Cartesian sections of $\mathcal{Z}'' \times_S (N(\Delta)^{op} \times \{ 1 \}) \to N(\Delta)^{op}$ is even isomorphic to $\mathcal{X}^{/U_{\bullet}}$, and I really don't see how. After writing things as explicitly as I can, it seems to me that objects in the former are functors $N(\Delta)^{op} \times N(\Delta)^{op} \times \Delta^1 \times \Delta^1 \to \mathcal{Z}$ + conditions, whereas objects in the latter are functors $N(\Delta)^{op} \times \Delta^1 \to \mathcal{Z}$ + conditions, and none of these conditions collapses any of those factors. What am I missing? Is there a formal way to obtain an isomorphism there?
- Just below this, at the beginning of Remark 6.3.5.17, it is claimed that we can identify $Shv_{\hat{\mathcal{S}}} (\mathcal{X})$ with $S^{-1} \mathcal{\hat{P}(C)}$ by using 5.1.5.6 and 5.5.4.20. What these directly lead to is that $Shv_{\mathcal{\hat{S}}}(\mathcal{X})$ can be identified with the full subcategory of $Fun^R (\mathcal{P(C)}^{op}, \mathcal{\hat{S}})$ spanned by the functors sending the morphisms in $S$ to equivalences in $\mathcal{\hat{S}}$. How should I conclude from there?
