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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?
Good question! I’ve thought about this before, but will have to recall what I was thinking. I only did a Cech cover, not a hypercover, though
Clearly a ’non-point-finite’ partition of unity for a covering map $p\colon U \to X$ is a function $\phi\colon U\to [0,1]$ with a fibrewise measure $d\mu(x)$ such that the pushforward $\int_{X} \phi d\mu(x)$ along $p$ is the constant function $X\to \{1\}\to [0,1]$. Then one can induce a cover refining $p$ by taking the pullback of $(0,1] \to [0,1]$ along $\phi$. It would be interesting to see if one could phrase this entirely in terms of (locally compact, say) topological groupoids with a chosen Haar measure.
The argument that one can improve to a locally finite partition of unity can be seen here (Google Books link: may or may not be visible), but I can’t immediately see how one would get an ’improvement’ of the above analogous to the usual case, without assume the existence of chosen local sections of $p$ (assuming they exist).
Yes, see Murray’s original paper.
EDIT: in more detail, let $\pi\colon Y\to M$ be a cover such that $Y$ admits partitions of unity. One can use a partition of unity on $Y^{[2]} = Y\times_M Y$ to get a connection $A$ on the bundle $p\colon E\to Y^{[2]}$, but then this might not be a bundle gerbe connection. The error is a 1-form $a$ on $Y^{[3]}$, which satisfies $\delta(a)=0$. By acyclicity of $(\Omega^n(Y^{[\bullet]}),\delta)$ this means that $a = \delta(b)$ for some 1-form $b$ on $Y{[2]}$. Then $A' = A - p^*b$ is a bundle gerbe connection. Then $\delta(dA') = 0$ (letting $dA'$ stand for the descended 2-form on $Y^{[2]}$), so $dA' = \delta(f)$ for a 2-form $f$ on $Y$, i.e. a curving.
I don’t know of any direct argument not using the acyclicity result.
Dmitri, if a tag is to be composed of more than one word, then please use some connective, say a dash. Otherwise it makes a lot of small tags for unwanted pieces of the phrase. For example here you introduced probably an unwanted tag “of”, alike it happened also in the entry https://nforum.ncatlab.org/discussion/6295/good-open-covers-and-partitions-of-unity-for-plmanifolds
@zskoda: Okay. Maybe the misleading prompt “Tags (Comma Separated)” should be changed to something more appropriate then, e.g., “Tags (Space or Comma Separated)”.
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