Cech cocycles for differential characteristic classes -- An infinity-Lie theoretic construction

Understand

What are called secondary characteristic classes in Chern-Weil theory are a refinement of ordinary characteristic classes of principal bundles from cohomology to differential cohomology.

• We consider the problem of refining the construction of secondary characteristic classes from cohomology sets to cocycle spaces; and from Lie groups to higher connected covers of Lie groups by smooth infinity-groups, i.e., by smooth groupal A-infinity-spaces.
• Namely, we realize differential characteristic classes as morphisms from infinity-groupoids of smooth principal infinity-bundles with connections to infinity-groupoids of higher U(1)-gerbes with connections.
• This allows us to study the homotopy fibers of the differential characteristic maps thus obtained and to show how these describe differential obstruction problems.