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Differential cohomology theories as sheaves of spectra (1311.3188v1)

Published 13 Nov 2013 in math.KT, math.AT, and math.CT

Abstract: We show that every sheaf on the site of smooth manifolds with values in a stable (infinity,1)-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 suggest 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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