Symmetric monoidal structure on non-commutative motives

In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Mot loc dg of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Mot loc dg in terms of non-connective algebraic K -theory; (2) a fully-faithful embedding of Kontsevich's category KMM k of non-commutative mixed motives into the base category Mot loc dg ( e ) of the localizing motivator; (3) a simple construction of the Chern character maps from non-connective algebraic K -theory to negative and periodic cyclic homology; (4) a precise connection between Toën's secondary K -theory and the Grothendieck ring of KMM k ; (5) a description of the Euler characteristic in KMM k in terms of Hochschild homology.