Differential cohomology theories as sheaves of spectra

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Abstract

We show that every sheaf on the site of smooth manifolds with values in a stable (Formula presented.) -category (like spectra or chain complexes) gives rise to a “differential cohomology diagram” and a homotopy formula, which are common features of all classical examples of differential cohomology theories. These structures are naturally derived from a canonical decomposition of a sheaf into a homotopy invariant part and a piece which has a trivial evaluation on a point. In the classical examples the latter is the contribution of differential forms. This decomposition suggests a natural scheme to analyse new sheaves by determining these pieces and the gluing data. We perform this analysis for a variety of classical and not so classical examples.

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Bunke, U., Nikolaus, T., & Völkl, M. (2016). Differential cohomology theories as sheaves of spectra. Journal of Homotopy and Related Structures, 11(1), 1–66. https://doi.org/10.1007/s40062-014-0092-5

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