\(\mathbf{A}^1\)-homotopy theory of noncommutative motives (Q893798)

Summary: In this article we continue the development of a theory of noncommutative motives, initiated in [the author, Duke Math. J. 145, No. 1, 121--206 (2008; Zbl 1166.18007)]. 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.