[Paper Review] The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)

Jack Morava|arXiv (Cornell University)|Aug 23, 2011

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper proposes that the cosmic Galois group—arising from motivic Galois theory—is Koszul dual to Waldhausen’s A(*)-theory spectrum, using derived categories of spectra and topological cyclic homology. It establishes a duality between the ring spectrum of topological K-theory and its Koszul dual via THH and TC, linking motivic structures to homotopy theory and revealing connections to quasisymmetric functions and odd zeta values.

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.

Motivation & Objective

• To explore the relationship between the cosmic Galois group and Waldhausen’s A(*)-theory in the context of motivic categories.
• To establish a Koszul duality framework between K-theory and topological cyclic homology (TC) spectra.
• To unify motivic structures in algebraic geometry and stable homotopy theory via spectral categories enriched in A(*)-theory.
• To investigate how the Betti realization of algebraic varieties fits into this motivic framework.
• To reconcile the motive of a variety with the stable homotopy type of its underlying topological space through rationalized TC and linearized motivic categories.

Proposed method

• Constructs a big spectral category of pre-motives enriched over Waldhausen’s A(*)-theory spectrum, using K-theory of exact functors between perfect ∞-categories.
• Defines a pre-triangulated completion of this category via Karoubi-Villamayor-style localization, generalizing Grothendieck’s construction of motives.
• Applies Koszul duality to the K-theory spectrum $K(\$)$, identifying its dual as $\mathrm{RHom}_{K(\$)}(\$, \$)$, which models the group algebra of the cosmic Galois group.
• Uses topological cyclic homology (TC) and THH to construct a cyclotomic variant of the motivic category, with morphism objects in $\$_\dagger{\rm TC}$-comodules.
• Introduces a linearized version of the TC-based motivic category, where rationalized morphism objects recover rational stable homotopy groups.
• Leverages the cyclotomic trace to relate $K(\$)$ to $TC(\$)$, identifying $TC(\$)$ with $\$∨ \Sigma \mathbb{C}P^\infty_{-1}$ at odd primes, enabling duality constructions.

Experimental results

Research questions

• RQ1 How can the cosmic Galois group be realized as the Koszul dual of Waldhausen’s A(*)-theory spectrum?
• RQ2 What is the role of topological cyclic homology (TC) in realizing motivic duality between K-theory and its dual?
• RQ3 How does the Betti realization of algebraic varieties relate to the motivic category built from A(*)-theory?
• RQ4 Can the rationalized motivic category derived from TC recover the rational stable homotopy category of finite spectra?
• RQ5 What is the significance of the appearance of quasisymmetric functions and odd zeta values in this duality framework?

Key findings

• The Koszul dual of $K(\$)$ is identified as $\$_\dagger{K(\$)} = \mathrm{RHom}_{K(\$)}(\$, \$)$, which models the group algebra of the cosmic Galois group.
• The rationalized version of the Koszul dual $\$_\dagger{K(\$)} \otimes \mathbb{Q}$ is isomorphic to the algebra of quasisymmetric functions over $\mathbb{Q}$, corresponding to a pro-unipotent group with free Lie algebra.
• The cyclotomic trace identifies $TC(\$)$ with $\$∨ \Sigma \mathbb{C}P^\infty_{-1}$ at regular odd primes, enabling a duality framework via THH and TC.
• The linearized motivic category $\mathsf{Mot}^{\rm lin}_{{\rm TC}_\dagger}$ has rationalized morphism objects isomorphic to $[Y, X]_{\mathbb{Q}}$, recovering the rational stable homotopy category.
• The construction suggests a coassembly map $\mathrm{TC}(DX) \to [X_+, \mathrm{TC}(\$)]$, linking homotopy fixed points of THH to TC via $\mathbb{T}$-equivariant structures.
• The $\mathbb{T}$-action on the Lie algebra of generators in odd degrees (from THH) mirrors Deligne’s motivic Galois group for mixed Tate motives, linking to arithmetic zeta values.