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

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 the- ory admits a unique extension to higher K-theory. As an application, the higher trace maps and the higher Chern characters originally con- structed by Dennis and Karoubi, respectively, can be obtained in an elegant, unified, and conceptual way from our general results.

[1]  Gonçalo Tabuada.  Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives , 2011, 1101.0731.

[2]  Denis-Charles Cisinski,et al.  Symmetric monoidal structure on non-commutative motives , 2010, Journal of K-Theory.

[3]  Denis-Charles Cisinski,et al.  Non-connective K-theory via universal invariants , 2009, Compositio Mathematica.

[4]  D. Kaledin,et al.  Motivic structures in non-commutative geometry , 2010, 1003.3210.

[5]  Gonçalo Tabuada.  A universal characterization of the Chern character maps , 2010, 1002.3726.

[6]  M. Kontsevich.  Notes on Motives in Finite Characteristic , 2007, math/0702206.

[7]  A. Neeman,et al.  Additivity for derivator K-theory , 2008 .

[8]  Gonçalo Tabuada.  Higher K-theory via universal invariants , 2007, 0706.2420.

[9]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[10]  A. Kuku,et al.  Higher Algebraic K-Theory , 2006 .

[11]  V. Drinfeld.  DG quotients of DG categories , 2002, math/0210114.

[12]  M. Bergh,et al.  Generators and representability of functors in commutative and noncommutative geometry , 2002, math/0204218.

[13]  N. Strickland,et al.  MODEL CATEGORIES (Mathematical Surveys and Monographs 63) , 2000 .

[14]  J. Loday.  Non-commutative Differential Geometry , 1998 .

[15]  John D. S. Jones,et al.  Bivariant cyclic theory , 1989 .

[16]  D. M. Kan,et al.  Homotopy Limits, Completions and Localizations , 1987 .

[17]  M. Karoubi.  Homologie cyclique et K-théorie , 1987 .

[18]  K. Herrmann.  Die mathematischen Hilfsmittel des Physikers. Von E. Madelung. Mit 20 Textfiguren (Het 4 der „Grundlehren der mathematischen Wissenschaften”.) Berlin, Verlag von Julius Springer, 1922 , 1923 .