In this article we continue the development of a theory of noncommutative motives, initiated in . We construct categories of -homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's -theory () and Weibel's homotopy -theory (). As an application, we obtain a complete classification of all the natural transformations defined on . This leads to a streamlined construction of Weibel's homotopy Chern character from to periodic cyclic homology. Along the way we extend Dwyer–Friedlander's étale -theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.
Cite this article
Gonçalo Tabuada, -homotopy theory of noncommutative motives. J. Noncommut. Geom. 9 (2015), no. 3, pp. 851–875DOI 10.4171/JNCG/210