A categorification of the Chern character

In this talk I will present joint work with Hoyois and Scherotzke on a categorification
of the Chern character, that refines earlier work of Toen and  Vezzosi and of Ganter
and Kapranov. If $X$ is an algebraic stack, our categorifed 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 $LX$.
Our construction has applications to Toen and Vezzosi's secondary Chern character.