Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism $\mathcal{U}_l$ from the pointed derivator $\mathsf{HO}(\mathsf{dgcat})$ associated with the Morita homotopy theory of dg categories to a triangulated strong derivator $\mathcal{M}_{\rm dg}^{\rm loc}$ such that $\mathcal{U}_l$ 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 $\mathcal{U}_a$ from $\mathsf{HO}(\mathsf{dgcat})$ to a strong triangulated derivator $\mathcal{M}_{\rm dg}^{\rm add}$ that satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen's $K$ -theory becomes corepresentable in the target of the universal additive invariant. This is the first conceptual characterization of Quillen and Waldhausen's $K$ -theory (see [34], [43]) since its definition in the early 1970s. As an application, we obtain for free the higher Chern characters from $K$ -theory to cyclic homology