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?

]]>Is anything known about the existence of partitions of unity and good open covers for PL-manifolds?

Here a good open cover of a PL-manifold is a locally finite open cover {U_i} such that every finite intersection of U_i is either empty or PL-isomorphic to R^n.

A partition of unity subordinate to an open cover {U_i} of a PL-manifold X is a family of nonnegative PL-functions f_i: X→R such that supp f_i is a subset of U_i, supp f_i form a locally finite family, and the sum of f_i is 1.

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