Higher K-theory via universal invariants

Abstract

Using the formalism of Grothendieck's derivators, we construct the universal localizing invariant of differential graded (dg) categories. By this we mean a morphism $U_l$ from the pointed derivator $\mathsf{HO}(\mathsf{dgcat})$ associated with the Morita homotopy theory of dg categories to a triangulated strong derivator $M_{\mathrm{dg}}^{\mathrm{loc}}$ such that $U_l$ 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 $U_a$ from $\mathsf{HO}(\mathsf{dgcat})$ to a strong triangulated derivator $M_{\mathrm{dg}}^{\mathrm{add}}$ 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