Norms in equivariant homotopy theory
Tobias Lenz, Sil Linskens, Phil Pützstück
TL;DR
The work establishes a comprehensive higher-categorical framework for multiplicative structures in equivariant and global stable homotopy theory. It proves that normed algebras in genuine $G$-spectra are equivalent to strictly commutative algebras in $G$-symmetric spectra, and it extends this equivalence to ultra-commutative global ring spectra via a global, parametrized algebraic approach. By developing distributivity, free functors, and Kan extensions in parametrized settings, the authors obtain monadicity results and universal descriptions of free normed algebras, both in equivariant and global contexts. They further show that global ultra-commutative ring spectra can be described as normed algebras in a global spectrum category and relate these to global Tambara functors, establishing a robust bridge between algebraic and spectral viewpoints. The paper concludes with a detailed globalization program, proving that equivariant multiplicative structures arise from a global normed-algebra framework and culminate in explicit equivalences that preserve forgetful functors and multiplicative refinements across finite groups and the global category.
Abstract
We show that the $\infty$-category of normed algebras in genuine $G$-spectra, as introduced by Bachmann-Hoyois, is modelled by strictly commutative algebras in $G$-symmetric spectra for any finite group $G$. We moreover provide an analogous description of Schwede's ultra-commutative global ring spectra in higher categorical terms. Using these new descriptions, we exhibit the $\infty$-category of ultra-commutative global ring spectra as a partially lax limit of the $\infty$-categories of genuine $G$-spectra for varying $G$, in analogy with the non-multiplicative comparison of Nardin, Pol, and the second author. Along the way, we establish various new results in parametrized higher algebra, which we hope to be of independent interest.