Normed equivariant ring spectra and higher Tambara functors

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 ∞-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 ∞-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.