Open Access

Homotopy-theoretically enriched categories of noncommutative motives

Research in the Mathematical Sciences20152:8

DOI: 10.1186/s40687-015-0028-7

Received: 14 March 2014

Accepted: 17 April 2015

Published: 10 June 2015


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