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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/6477