A simple criterion for extending natural transformations to higher $K$-theory

Abstract

In this article we introduce a very simple an widely applicable criterion for extending natural transformations to higher $K$-theory. More precisely, we prove that every natural transformation defined on the Grothendieck group and with values in an additive theory admits a unique extension to higher $K$-theory. As an application, the higher trace maps and the higher Chern characters originally constructed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.