Differential cohomology theories as sheaves of spectra

Understand

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.