${\mathbf{A}}^1$-homotopy theory of noncommutative motives

Abstract

In this article we continue the development of a theory of noncommutative motives, initiated in [30]. We construct categories of ${\mathbf{A}}^1$-homotopy noncommutative motives, describe their universal properties, and compute their spectra of morphisms in terms of Karoubi–Villamayor's $K$-theory ($KV$) and Weibel's homotopy $K$-theory ($KH$). As an application, we obtain a complete classification of all the natural transformations defined on $KV,KH$. This leads to a streamlined construction of Weibel's homotopy Chern character from $KV$ to periodic cyclic homology. Along the way we extend Dwyer–Friedlander's étale $K$-theory to the noncommutative world, and develop the universal procedure of forcing a functor to preserve filtered homotopy colimits.