  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 $S^0$.
