  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, hence higher gauge fields.
We discuss various models of the axioms and applications to fundamental notions
and constructions in quantum 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 hierarchies of
higher dimensional Chern-Simons-type field theories, their higher
Wess-Zumino-Witten-type boundary field theories and all further higher
codimension defect field theories. We close with an outlook on the
cohomological quantization of such higher boundary prequantum field theories by
a kind of cohesive motives.
