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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 11th 2015
    • (edited Sep 11th 2015)

    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?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 11th 2015

    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

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 14th 2015

    Clearly a ’non-point-finite’ partition of unity for a covering map p:UXp\colon U \to X is a function ϕ:U[0,1]\phi\colon U\to [0,1] with a fibrewise measure dμ(x)d\mu(x) such that the pushforward Xϕdμ(x)\int_{X} \phi d\mu(x) along pp is the constant function X{1}[0,1]X\to \{1\}\to [0,1]. Then one can induce a cover refining pp by taking the pullback of (0,1][0,1](0,1] \to [0,1] along ϕ\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 pp (assuming they exist).

    • CommentRowNumber4.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 15th 2015
    Any hypercover H→X can be presented as a sequence of ordinary covers H_n→M_n(H),
    where M_n denotes the nth matching object.
    Consequently, one could define a partition of unity for a hypercover as
    a sequence of partitions of unity for H_n→M_n(H).
    The question now is whether we can use this data to construct
    a connection on a bundle gerbe.
    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 15th 2015
    • (edited Sep 15th 2015)

    Yes, see Murray’s original paper.

    EDIT: in more detail, let π:YM\pi\colon Y\to M be a cover such that YY admits partitions of unity. One can use a partition of unity on Y [2]=Y× MYY^{[2]} = Y\times_M Y to get a connection AA on the bundle p:EY [2]p\colon E\to Y^{[2]}, but then this might not be a bundle gerbe connection. The error is a 1-form aa on Y [3]Y^{[3]}, which satisfies δ(a)=0\delta(a)=0. By acyclicity of (Ω n(Y []),δ)(\Omega^n(Y^{[\bullet]}),\delta) this means that a=δ(b)a = \delta(b) for some 1-form bb on Y[2]Y{[2]}. Then A=Ap *bA' = A - p^*b is a bundle gerbe connection. Then δ(dA)=0\delta(dA') = 0 (letting dAdA' stand for the descended 2-form on Y [2]Y^{[2]}), so dA=δ(f)dA' = \delta(f) for a 2-form ff on YY, i.e. a curving.

    I don’t know of any direct argument not using the acyclicity result.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeSep 17th 2015
    Interesting. I guess a partition of unity for a hypercover should be able to construct A' more directly.
    • CommentRowNumber7.
    • CommentAuthorzskoda
    • CommentTimeFeb 10th 2016

    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

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeFeb 20th 2016

    @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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