  We extend the Chern character on K-theory, in its generalization to the
Chern-Dold character on generalized cohomology theories, further to (twisted,
differential) non-abelian cohomology theories, where its target is a
non-abelian de Rham cohomology of twisted L-infinity algebra valued
differential forms. The construction amounts to leveraging the fundamental
theorem of dg-algebraic rational homotopy theory to a twisted non-abelian
generalization of the de Rham theorem. We show that the non-abelian character
reproduces, besides the Chern-Dold character, also the Chern-Weil homomorphism
as well as its secondary Cheeger-Simons homomorphism on (differential)
non-abelian cohomology in degree 1, represented by principal bundles (with
connection); and thus generalizes all these to higher (twisted, differential)
non-abelian cohomology, represented by higher bundles/higher gerbes (with
higher connections). As a fundamental example, we discuss the twisted
non-abelian character map on twistorial Cohomotopy theory over 8-manifolds,
which can be viewed as a twisted non-abelian enhancement of topological modular
forms (tmf) in degree 4. This turns out to exhibit a list of subtle topological
relations that in high energy physics are thought to govern the charge
quantization of fluxes in M-theory.
