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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 -groups, i.e., by smooth groupal A spaces. Namely, we realize differential characteristic classes as morphisms from -groupoids of smooth principal -bundles with connections to -groupoids of higher U (1)-gerbes with connections. This allows us to
Fiorenza et al. (Sun,) studied this question.