Differential cohomology theories as sheaves of spectra on the site of manifolds

Abstract: Generalized cohomology theories can be described by spectra. One approach to differential cohomology theories defines them as ‘smooth spectra’: sheaves on manifolds with values in spectra. This abstract categorical definition allows for a general ‘recollement’ theorem, which tells one precisely how to build a differential cohomology theory: you need a spectrum, certain geometric data and a ‘way to mix the two’. It can then be shown that every such differential cohomology theory fits a certain hexagon diagram similar to what we have seen in the first talk. We then discuss examples, in particular recovering Deligne cohomology.