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?

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