  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. This applies
in particular to the higher twisted differential spin structures called twisted
differential string structures and twisted differential fivebrane structures.
