Abstract
Waldhausen’s K-theory of the sphere spectrum (closely related to the algebraic K-theory of the integers) is naturally augmented as an 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 . This paper argues that the rationalizations of categories of noncommutative motives defined recently by Blumberg, Gepner, and Tabuada consequently have natural enrichments, with morphism objects in the derived category of mixed Tate motives over
. We suggest that homotopic descent theory lifts this structure to define a category of motives defined not over
but over the sphere ring-spectrum S
0.
Mathematics subject classification: 11G, 19F, 57R, 81T
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Acknowledgements
I am deeply indebted to Andrew Blumberg, Kathryn Hess, and Nitu Kitchloo for help and encouragement in the early stages of this work; and to Andrew Baker, Birgit Richter, and John Rognes for their advice and intervention in its later stages. Thanks to all of them - and to some very perceptive and helpful referees - for their interest and patience. Any mistakes, misunderstandings, and oversimplifications are my responsibility.
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Morava, J. Homotopy-theoretically enriched categories of noncommutative motives. Mathematical Sciences 2, 8 (2015). https://doi.org/10.1186/s40687-015-0028-7
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DOI: https://doi.org/10.1186/s40687-015-0028-7