Symmetric monoidal structure on Non-commutative motives

TL;DR

This paper develops a symmetric monoidal structure on the localizing motivator of dg categories, enabling new computations and embeddings in non-commutative motives, with applications to algebraic K-theory and cyclic homology.

Contribution

It constructs a symmetric monoidal structure on the localizing motivator of dg categories and applies it to various problems in non-commutative motives.

Findings

Computed spectra of morphisms in Mot via non-connective algebraic K-theory

Embedded Kontsevich's non-commutative mixed motives into Mot(e)

Constructed Chern character maps to cyclic homology

Abstract

In this article we further the study of non-commutative motives. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Mot of dg categories. As an application, we obtain : (1) a computation of the spectra of morphisms in Mot in terms of non-connective algebraic K-theory; (2) a fully-faithful embedding of Kontsevich's category KMM of non-commutative mixed motives into the base category Mot(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 KMM ; (5) a description of the Euler characteristic in KMM in terms of Hochschild homology.