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 Ul from the pointed derivator HO(dgcat) associated with the Morita homotopy theory of dg categories to a triangulated strong derivator Mdgloc such that Ul 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 Ua from HO(dgcat) to a strong triangulated derivator Mdgadd 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