The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)
arXiv:1108.4627
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.
Notes from a talk at the Hamburg August 2011 conference http://www.math.uni-hamburg.de/home/richter/hh2011.html on structured ring spectra
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