Normed equivariant ring spectra and higher Tambara functors

TL;DR

This paper extends equivariant infinite loop space theory to include multiplicative norms, providing a new description of connective normed equivariant ring spectra as space-valued Tambara functors, and connecting these to algebraic K-theory spectra.

Contribution

It introduces a homotopy-coherent notion of normed rings, identifies space-valued Tambara functors with normed algebras, and relates these to the Hill-Hopkins-Ravenel norms in equivariant spectra.

Findings

Constructed a multiplicative refinement of the comparison between genuine G-spectra and Mackey functors.

Identified space-valued Tambara functors with normed algebras in a new normed monoidal structure.

Produced normed ring structures on equivariant algebraic K-theory spectra.

Abstract

In this paper we extend equivariant infinite loop space theory to take into account multiplicative norms: For every finite group $G$, we construct a multiplicative refinement of the comparison between the $\infty$-categories of connective genuine $G$-spectra and space-valued Mackey functors, first proven by Guillou-May, and use this to give a description of connective normed equivariant ring spectra as space-valued Tambara functors. In more detail, we first introduce and study a general notion of homotopy-coherent normed (semi)rings, and identify these with product-preserving functors out of a corresponding $\infty$-category of bispans. In the equivariant setting, this identifies space-valued Tambara functors with normed algebras with respect to a certain normed monoidal structure on grouplike $G$-commutative monoids in spaces. We then show that the latter is canonically equivalent to the normed monoidal structure on connective $G$-spectra given by the Hill-Hopkins-Ravenel norms. Combining our comparison with results of Elmanto-Haugseng and Barwick-Glasman-Mathew-Nikolaus, we produce normed ring structures on equivariant algebraic K-theory spectra.