Symmetric monoidal structure on non-commutative motives

Abstract. In this article we further the study of non-commutative motives, initiated in [11, 42]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motloc of dg categories. As dg an application, we obtain: (1) a computation of the spectra of morphisms in Motloc in terms of non-connective algebraic K-theory; (2) a fully-faithful dg embedding of Kontsevich’s category KMMk of non-commutative mixed mo- tives into the base category Motloc(e) of the localizing motivator; (3) a sim- dg ple construction of the Chern character maps from non-connective algebraic K-theory to negative and periodic cyclic homology; (4) a precise connection between To ̈en’s secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild ho- mology.