Higher K-theory via universal invariants

TL;DR

This paper constructs universal invariants for dg categories using derivators, providing a conceptual framework that characterizes K-theory and enables deriving higher Chern characters to cyclic homology.

Contribution

It introduces the universal localizing and additive invariants of dg categories via derivators, offering a new conceptual understanding of K-theory and related invariants.

Findings

Waldhausen K-theory is realized as a mapping space in the universal additive invariant

The construction provides a universal property characterizing K-theory

Higher Chern characters to cyclic homology are obtained for free

Abstract

Using the formalism of Grothendieck's derivators, we construct `the universal localizing invariant of dg categories'. By this, we mean a morphism U_l from the pointed derivator associated with the Morita homotopy theory of dg categories to a triangulated strong derivator M^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. Similary, we construct the `the universal additive invariant of dg categories', i.e. the universal morphism of derivators U_a to a strong triangulated derivator M^add which satisfies the first two properties but the third one only for split exact sequences. We prove that Waldhausen K-theory appears as a mapping space in the target of the universal additive invariant. This is the first conceptual characterization of Quillen-Waldhausen's K-theory since its definition in the early 70's. As an application we obtain for free the higher Chern characters from K-theory to cyclic homology.