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The nLab entry Cech cohomology claims in its first sentence that Cech cohomology of a site $C$ is the cohomology of the $(\infty,1)$-topos of presheaves on $C$ localized at Cech covers. I’m having trouble reconciling this with Remark 7.2.2.17 of Higher Topos Theory, which claims that the cohomology of this topos is ordinary sheaf cohomology. I used to assume the claim the nLab makes without thinking about it (it’s pretty plausible…), but now I think it may be wrong. Specifically, Eilenberg-Mac Lane objects are truncated and therefore hypercomplete, so they automatically satisfy descent with respect to all hypercovers, and so cohomology of an $(\infty,1)$-topos and of its hypercompletion should always be the same.
So, is the $(\infty,1)$-topos referred to in the nLab page something different than the topos of sheaves defined by Lurie? Or is there no way to view Cech cohomology as the cohomology of an $(\infty,1)$-topos?
That was indeed a wrong statement on the $n$Lab page. The cohomology in the topos is always the correct sheaf cohomology, and Cech cohomology is an algorithm for computing it, which may or may not apply.
Cech cohomology really comes in when working with a local projective model structure on the simplicial presheaves over a site, where lots of objects are fibrant. Then computing the “derived hom” (e.g. def. 7.2.2.14 in HTT) comes down to finding a cofibrant replacement of the object that we compute the cohomology of. It then may or may not happen that the Cech nerves of covering families are split hypercovers and hence cofibrant in the projective model structure. If so, Cech cohomology computes the correct derived hom. If not, then not.
Thanks for the confirmation! It cleared up a lot of things.
It seems to me that the key difference between hypercovers and Cech covers (say in the world of simplicial presheaves) is in how much these classes differ from their saturation (where the saturation of $C$ is the class of all maps that become isomorphism after you invert all morphisms in $C$). The class $HC$ of hypercovers is close enough to being saturated: among all $HC^{-1}$-isomorphisms into a representable, there are enough hypercovers. On the other hand, the class $CC$ of Cech covers is less close in general: among all $CC^{-1}$-isomorphisms into a representable, there may not be enough Cech covers.
Do you happen to know how to compute mapping spaces in the Cech-localized topos? Say $Map(X,F)$ where $X$ is representable and $F$ is locally fibrant. If $F$ is truncated, it’s the same as in the hypercomplete topos, so we know we can resolve $X$ by bounded hypercovers. I wonder if bounded hypercovers would also give the correct answer for non-truncated $F$. Dugger-Hollander-Isaksen and the $n$Lab page do not address this question, so I guess it may be unknown.
Once $F$ is locally fibrant, that question does not depend on $F$ anymore. All you need is a cofibrant resolution of $X$. (Am I misunderstanding your question?)
Locally fibrant is much weaker than fibrant, so whatever you replace $X$ with may not give you the correct mapping space. I’m referring to the Verdier hypercovering theorem which says that you can still compute the mapping space, without resolving $F$, by taking the colimit of the simplicial sets of maps over all hypercovers. This is the most useful tool to compute mapping spaces in the hypercomplete topos, because locally fibrant is an essentially empty condition. See DHI, Theorem 7.6 (b) or Jardine for the statement of the theorem. That’s what I meant when I said that there are enough hypercovers mapping into a given object. Of course Verdier’s theorem holds for abstract reasons if you take that colimit over enough weak equivalences to $X$, but I wonder what class of hypercovers to use (or equivalently, what hypercovers to remove) to get the mapping space in the Cech-localized topos. The only obvious candidate is the class of bounded hypercovers, but DHI doesn’t go there. Without such a theorem the Cech-localization seems uncomputable in practice (whenever it differs from its hypercompletion, that is).
For example, Verdier’s theorem is what gives us a relatively simple point-set model for the shape of the hypercompletion of the $(\infty,1)$-topos of sheaves on a locally connected $1$-site (the Artin-Mazur stuff).
Locally fibrant is much weaker than fibrant,
Oh, you didn’t mean fibrant in the local model structure. You meant stalkwise fibrant. Sure.
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