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Started the page cd-structure.
A minor thing, but it’s not true in general that the (∞,1)-topos of sheaves for a cd-structure is hypercomplete. The (∞,1)-topos of a coherent topological space is always given by a cd-structure but it may not be hypercomplete (see counterexample 6.5.4.5 in HTT).
Thanks for that correction. I guess I was misled by the fact that, for noetherian schemes, the Nisnevich topology is generated by a cd-structure, and the associated topos is hypercomplete, while both of these facts are false in general (right?).
Do you know any way to characterize the toposes that arise from cd-structures? I guess a necessary condition is generation under contractible colimits by the Yoneda embedding, right?
I don’t know any sufficient conditions. Another fairly strong necessary condition is that representables are compact in the (∞,1)-topos of sheaves.
The Nisnevich topology on non-noetherian schemes is always a confusing topic. Let me try to shed some light on it. Originally, Nisnevich defined the following pretopology: $\{f_i: X_i\to X\}$ is a covering if each $f_i$ is étale and if every point $x\in X$ has a preimage in some $X_i$ with the same residue field. Let’s call this topology $Nis_{bad}$. In DAG XI, Lurie has a slightly different definition of the Nisnevich topology on affine schemes, which can be extended to all schemes by throwing in the Zariski topology; let’s call this topology $Nis$. So we have
$Zar \leq Nis \leq Nis_{bad} \leq et.$Now let’s fix a scheme $X$ and consider, for each of these topologies $\tau$, the (∞,1)-topos $X_\tau$ of $\tau$-sheaves on the category $Et_X$ of étale $X$-schemes (that’s a large category but nevertheless you get a topos). Then:
For these reasons, $Nis$ is often more appropriate than $Nis_{bad}$. But $Nis_{bad}$ isn’t all that bad. For example, for any $X$, the (∞,1)-topos $X_{Nis_{bad}}^\wedge$ has a conservative family of points given by the henselizations of the local rings of étale $X$-schemes. For $X$ qcqs, $X_{Nis}^{\wedge}$ also has enough points by Lurie’s generalization of Deligne’s theorem, but I don’t know a nice description of the points.
I changed a bit the definition of the associated topology, because what was defined before was not really a pretopology.
I also upgraded Prop. 1 to (∞,1)-presheaves instead of just presheaves, and added a reference.
That’s very helpful, thanks!
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