Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator such that commutes with filtered homotopy colimits, preserves the point, sends each exact sequence of dg categories to a triangle, and is universal for these properties.
Similarly, we construct the universal additive invariant of dg categories, that is, the universal morphism of derivators from to a strong triangulated derivator that satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen's -theory becomes corepresentable in the target of the universal additive invariant. This is the first conceptual characterization of Quillen and Waldhausen's -theory (see , ) since its definition in the early 1970s. As an application, we obtain for free the higher Chern characters from -theory to cyclic homology
"Higher -theory via universal invariants." Duke Math. J. 145 (1) 121 - 206, 1 October 2008. https://doi.org/10.1215/00127094-2008-049