Partitions of unity for an ordinary cover can be used to write down explicit coboundaries and cocycles for geometric objects specified locally on an open cover.

Suppose now we have a geometric object specified using a hypercover, e.g., a bundle gerbe. Is there an analog of the usual notion of partition of unity that allows us to write down explicit formulas in a similar fashion, e.g., as in the article partitions of unity? For example, can one construct a connection on a bundle gerbe in a similar fashion as in the article connection on a bundle?

]]>The article local model structure on simplicial presheaves states that for a site with enough points stalkwise weak equivalences of simplicial presheaves coincide with weak equivalences of simplicial presheaves in the Bousfield localization of componentwise weak equivalences with respect to all hypercovers (i.e., weak equivalences in a local model structure).

Is a proof of this statement written up somewhere? (The article cited above gives a reference to Jardine, which claims, but does not prove this statement.)

Also, is it possible to formulate an analog of this statement for sites that do not have enough points? (Presumably we would have to talk about sufficiently refined (hyper)covers instead of points.)

]]>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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