This is an article in preparation
in the context of differential cohomology in a cohesive topos.
Abstract Using higher Chern-Weil theory of smooth ∞-groupoids we study systematically and in detail differential refinements of “topological structures” given by lifts of structure groups, such as notably the sequence orientation-, spin-, spin-, fivebrane-structure The resulting twisted differential structures in nonabelian differential cohomology control the target space differential cohomology of higher supersymmetric sigma-models, such as their quantum anomaly cancellation but also their duality structure, such as T-duality. Our general abstract approach unifies models that have been proposed in the literature and lifts them to higher categorical degree. Our concrete models provide local differential form data that reproduces equations as known in the string theory literature and exhibits them as special cases of a more general phenomenon: higher Chern-Weil theory.