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1. • CommentRowNumber1.
   • CommentAuthoradeelkh
   • CommentTimeOct 20th 2014
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   Author: adeelkh
   Format: MarkdownItexStarted the page [[cd-structure]].

   Started the page cd-structure.

2. • CommentRowNumber2.
   • CommentAuthorMarc Hoyois
   • CommentTimeOct 21st 2014
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   Author: Marc Hoyois
   Format: MarkdownItexA 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).

   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).

3. • CommentRowNumber3.
   • CommentAuthoradeelkh
   • CommentTimeOct 21st 2014
   • (edited Oct 21st 2014)
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   Author: adeelkh
   Format: MarkdownItexThanks 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorMarc Hoyois
   • CommentTimeOct 21st 2014
   • (edited Oct 21st 2014)
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   Author: Marc Hoyois
   Format: MarkdownItexI 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: - $X_{et}\subset X_{Nis_{bad}} \subset X_{Nis} \subset X_{Zar}$ - If $X$ is locally noetherian, $X_{Nis_{bad}}=X_{Nis}$. In general they are different. - If $X$ is qcqs, then $X_{Zar}$, $X_{Nis}$, and $X_{et}$ are coherent. In general, $X_{Nis_{bad}}$ isn't. - If $X=\lim_\alpha X^\alpha$ where the $X^\alpha$'s are qcqs, then $X_\tau=\lim_\alpha X^\alpha_\tau$ for $\tau=Zar,Nis,et$. Not so for $\tau=Nis_{bad}$. - If $X$ is qcqs, then $X_{Zar}$ and $X_{Nis}$ can be described as presheaves on finitely presented étale $X$-schemes satisfying a Mayer-Vietoris condition. Not so for $X_{Nis_{bad}}$ or $X_{et}$. - (Corollary) If $X$ is qcqs, then $X_{Zar}$ and $X_{Nis}$ are compactly generated (by finitely presented étale $X$-schemes). In general, $X_{Nis_{bad}}$ and $X_{et}$ aren't. - If $X$ is noetherian *and* of finite Krull dimension, then $X_{Zar}$ and $X_{Nis}=X_{Nis_{bad}}$ are hypercomplete. 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 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:

   • $X_{et}\subset X_{Nis_{bad}} \subset X_{Nis} \subset X_{Zar}$
   • If $X$ is locally noetherian, $X_{Nis_{bad}}=X_{Nis}$. In general they are different.
   • If $X$ is qcqs, then $X_{Zar}$, $X_{Nis}$, and $X_{et}$ are coherent. In general, $X_{Nis_{bad}}$ isn’t.
   • If $X=\lim_\alpha X^\alpha$ where the $X^\alpha$’s are qcqs, then $X_\tau=\lim_\alpha X^\alpha_\tau$ for $\tau=Zar,Nis,et$. Not so for $\tau=Nis_{bad}$.
   • If $X$ is qcqs, then $X_{Zar}$ and $X_{Nis}$ can be described as presheaves on finitely presented étale $X$-schemes satisfying a Mayer-Vietoris condition. Not so for $X_{Nis_{bad}}$ or $X_{et}$.
   • (Corollary) If $X$ is qcqs, then $X_{Zar}$ and $X_{Nis}$ are compactly generated (by finitely presented étale $X$-schemes). In general, $X_{Nis_{bad}}$ and $X_{et}$ aren’t.
   • If $X$ is noetherian and of finite Krull dimension, then $X_{Zar}$ and $X_{Nis}=X_{Nis_{bad}}$ are hypercomplete.

   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.

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeOct 21st 2014
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   Author: Marc Hoyois
   Format: MarkdownItexI 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.

   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.

6. • CommentRowNumber6.
   • CommentAuthoradeelkh
   • CommentTimeOct 22nd 2014
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   Author: adeelkh
   Format: MarkdownItexThat's very helpful, thanks!

   That’s very helpful, thanks!

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJun 26th 2022
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   Author: Urs
   Format: MarkdownItexhave expanded the Idea-section a bit, boosted the references a little, cross-linked with the new entry *[[Brown-Gersten property]]* made *completely decomposable Grothendieck topology* and variants redirect here <a href="https://ncatlab.org/nlab/revision/diff/cd-structure/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/cd-structure/12">v12</a>, <a href="https://ncatlab.org/nlab/show/cd-structure">current</a>

   have expanded the Idea-section a bit,

   boosted the references a little,

   cross-linked with the new entry Brown-Gersten property

   made completely decomposable Grothendieck topology and variants redirect here

   diff, v12, current

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 27th 2022
   • (edited Jun 27th 2022)
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   Author: DavidRoberts
   Format: MarkdownItexThe Zariski topology is given as an example, but not particularly explained. I'll add a little bit

   The Zariski topology is given as an example, but not particularly explained. I’ll add a little bit

9. • CommentRowNumber9.
   • CommentAuthorDavidRoberts
   • CommentTimeJun 27th 2022
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   Author: DavidRoberts
   Format: MarkdownItexAdded a description of the squares that define the Zariski topology. I guess one should get other Grothendieck topologies defined by classes of finite open covers in the same fashion, but I haven't added these. <a href="https://ncatlab.org/nlab/revision/diff/cd-structure/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/cd-structure/13">v13</a>, <a href="https://ncatlab.org/nlab/show/cd-structure">current</a>

   Added a description of the squares that define the Zariski topology.

   I guess one should get other Grothendieck topologies defined by classes of finite open covers in the same fashion, but I haven’t added these.

   diff, v13, current