Parametrised noncommutative motives and equivariant cubical descent in algebraic K-theory

TL;DR

This paper develops a parametrised noncommutative motives framework to enhance equivariant algebraic K-theory, introducing a cubical descent theory that refines K-theory spectra with E-infinity structures for finite 2-groups.

Contribution

It introduces parametrised perfect-stable categories and a cubical descent theory, extending noncommutative motives and providing new structures for equivariant algebraic K-theory.

Findings

Parametrised noncommutative motives corepresent algebraic K-theory.

Develops a cubical descent theory generalising equivariant Goodwillie calculus.

Refines equivariant K-theory spectra to E-infinity ring spectra with multiplicative norms.

Abstract

For an atomic orbital base category in the sense of Barwick-Dotto-Glasman-Nardin-Shah, we introduce the category of parametrised perfect-stable categories and use it to construct the parametrised version of noncommutative motives in which algebraic K-theory is corepresented. Furthermore, we initiate a rudimentary theory of parametrised cubes which could be of independent interest, generalising some of the elements in Dotto's theory of equivariant Goodwillie calculus beyond the equivariant case. Using this cubical theory, we show that in the equivariant case for finite 2-groups G, the parametrised noncommutative motives canonically refine to G-symmetric monoidal categories. Consequently, this endows the equivariant algebraic K-theory spectra for these groups with the structure of E-infinity-ring spectra equipped with multiplicative norms in the sense of Hill-Hopkins-Ravenel. Along the way, we will also provide a machine to manufacture G-symmetric monoidal categories from symmetric monoidal categories equipped with G-actions and elucidate how the aforementioned parametrised perfect-stable categories relate to Mackey functors valued in perfect-stable categories.