Homotopy-theoretically enriched categories of noncommutative motives

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 Z. 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 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 .

Topological invariants of some chemical reaction networks

“…These groups are torsion-free, and it will be convenient below to work with their characteristic zero localizations, defined by tensoring with Q. The completed localization B((b −1 )) (with b = b 1 [31]) can be regarded as a Z/2Z-graded filtered version of B * . §III The Baker-Richter spectrum This section summarizes some of the work of Baker and Richter on quasi-and noncommutative symmetric functions, and their role in algebraic topology.…”

The stable symplectic category and a conjecture of Kontsevich

“…One consequence of deep work of Hatcher, Waldhausen, Bökstedt, Rognes and others is a rational equivalence [21]( §3) (S ∨ΣkO) Q −→ K(S) Q of spectra, which yields a canonical identification Remarkably enough, these multizeta values play an important role in Connes,Kreimer, and Marcolli's Galois-theoretic reinterpretation [10,11] of the classical BPS renormalization theory of Feynman integrals, in which MZVs appear ubiquitously in explicit computations. Kontsevich found an action [19][ §4.6 Th 9] of the abelianization of GT Q on a moduli space for deformation quantizations of Poisson manifolds, through an action of the little disk operad on Hochschild homology.…”

Renormalization groupoids in algebraic topology

“…for the antipode of N. According to Ditters' conjecture, Q * is a free commutative algebra; over Q it is polynomial, generated by elements of degree 2 As in 2.1, the grading on N * can be encoded by adjoining a central unit Z0 of degree zero 2n indexed by Lyndon words of degree n, with multiplicative structure defined by a certain shuffle product on the basis {m I }. The quasisymmetric functions have interesting relations with Koszul duality [24]( §3) and number theory [6]( §4.4), but these will not be pursued here. The surjection…”