  We propose a categorification of the Chern character that refines earlier
work of To\"en and Vezzosi and of Ganter and Kapranov. If X is an algebraic
stack, our categorified Chern character is a symmetric monoidal functor from a
category of mixed noncommutative motives over X, which we introduce, to
S1-equivariant perfect complexes on the derived free loop stack LX. As an
application of the theory, we show that To\"en and Vezzosi's secondary Chern
character factors through secondary K-theory. Our techniques depend on a
careful investigation of the functoriality of traces in symmetric monoidal
(infinity,n)-categories, which is of independent interest.
