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In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of Čech covers.

Blockquote

The Nisnevich localization at all hypercovers is equivalent to the localization at Čech covers. But the etale localization doesn't have this property. A related post here gives some insights about "hypercompletion" from $\infty$-category point of view.

To show that Zariski or Nisnevich hyperdescent is the same as their Čech descent, one either uses the results of Brown-Bersten and Voevodsky or shows they have finite homotopy dimension in the higher topos setting.
When I first read about hypercovers, I thought people showed that Čech localization is the same as localization of all hypercovers under some conditions and then went on to reduce Čech decent condition to some simpler excision conditions for some topologies. Is there a way to show that the two localizations are the same for Nisnevich or Zariski from the properties listed above?

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  • $\begingroup$ Sheaves of sets are the local objects for localization at stalk-wise isomorphisms. Hypersheaves of spaces are the local objects for localization at stalk-wise equivalences. Thus comparing sheaves to hypersheaves is about looking at stalk-wise equivalences between sheaves and asking if they must be global equivalences. The original Brown-Gersten paper (cited by DHI) predates this language, but it's pretty clear that's what they are doing (for the Zariski topology). $\endgroup$ – Ben Wieland Aug 20 '19 at 21:52
  • $\begingroup$ For the negative direction, just write down a stalk-wise equivalence and check that both objects are sheaves. But if you only care about hypsh, why bother writing a counterex for the Galois top? For decades, they they hypsh-fied and proved relevant lemmas (eg, Thomason proved that the etale-sh of $K$-theory is already hyper.) DHI gives an example with an artificial site. But then Lurie came along and said that he cares about the category of sheaves, not just hypersheaves. He gave the example of the dualizing sheaf of the Hilbert cube $D_Q(U)=H(Q,Q-U)$, which is very similar to the DHI example. $\endgroup$ – Ben Wieland Aug 20 '19 at 22:02
  • $\begingroup$ Hi Ben, where did Thomason prove that? I'm surprised because on my reading Thomason never considered etale sheafification, only hypersheafifcation. Also, I think the only proof I know of this fact (when it holds true) uses the full Bloch-Kato conjecture, not just the Merkurjev-Suslin theorem as Thomason had access to. $\endgroup$ – Dustin Clausen Aug 21 '19 at 10:16
  • $\begingroup$ By the way, we give some answers to this question in the paper arxiv.org/abs/1905.06611, building on much previous work such as that mentioned by Ben (Brown-Gersten, Thomason). The upshot is that under finite Krull dimension hypothesis, there is no difference between descent and hyperdescent in the Zariski and Nisnevich settings, but no matter what finite dimension hypotheses you want to impose there is a difference in the etale setting. However all etale sheaves somehow related algebraic K-theory end up being hypersheaves anyway, under reasonable finite dimension hypotheses. $\endgroup$ – Dustin Clausen Aug 21 '19 at 10:21
  • $\begingroup$ @DustinClausen I don't see it in either Thomason or Mitchell, but Thomason did consider Čech cohomology, which is close to sheafification. $\endgroup$ – Ben Wieland Aug 22 '19 at 1:32

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