Differential cohomology in a cohesive ∞-topos

We formulate differential cohomology and Chern-Weil theory the theory of connections on fiber bundles and of gauge fields abstractly in the context of a certain class of higher toposes that we call cohesive. Cocycles in this differential cohomology classify higher principal bundles equipped with cohesive structure (topological, smooth, synthetic differential, supergeometric, etc.) and equipped with connections. We discuss various models of the axioms and wealth of applications revolving around fundamental notions and constructions in prequantum field theory and string theory. In particular we show that the cohesive and differential refinement of universal characteristic cocycles constitutes a higher Chern-Weil homomorphism refined from secondary caracteristic classes to morphisms of higher moduli stacks of higher gauge fields, and at the same time constitutes extended geometric prequantization – in the sense of extended/multi-tiered quantum field theory – of higher dimensional Chern-Simons-type field theories and Wess-Zumino-Witten-type field theories. This document, and accompanying material, is kept online at ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos