The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)

Abstract: K. Hess's theory of homotopical descent, applied to the large categories of motives defined recently by Blumberg, Gepner, and Tabuada, suggests that the Koszul dual of Waldhausen's K-theory of the sphere spectrum, regarded as a supplemented algebra via the Dennis trace, plays a very general role as a kind of motivic group. After tensoring with the rationals, the resulting Hopf algebra has close relations to the ring of quasi-symmetric functions and work of Baker and Richter on one hand, and on the other to work of Deligne and others on the motivic group for mixed Tate motives.