Higher traces, noncommutative motives, and the categorified Chern character

We propose a categorification of the Chern character that refines earlier work of Tan and Vezzosi and of Canter 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 S-1-equivariant perfect complexes on the derived free loop stack LX. As an application of the theory, we show that Tan 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. (C) 2017 Elsevier Inc. All rights reserved.