Homotopy-theoretically enriched categories of noncommutative motives

Waldhausen's $K$-theory of the sphere spectrum (closely related to the algebraic $K$-theory of the integers) is a naturally augmented $S^0$-algebra, and so has a Koszul dual. Classic work of Deligne and Goncharov implies an identification of the rationalization of this (covariant) dual with the Hopf algebra of functions on the motivic group for their category of mixed Tate motives over $\Z$. This paper argues that the rationalizations of categories of non-commutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over $\Z$. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over $\Z$ but over the sphere ring-spectrum $S^0$.Comment: An attempt at a more readable version. Some reshuffling, a few new references, small notational changes. Thanks to many for comments about foolish blunders and obscuritie