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(dg)(loc) of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Mot(dg)(loc) in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMMk of non-commutative mixed motives into the base category Mot(dg)(loc) (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 Toen's secondary K-theory and the Grothendieck ring of KMMk; (5) a description of the Euler characteristic in KMMk in terms of Hochschild homology.