Norms in equivariant homotopy theory

Abstract We show that the $\infty $ -category of normed algebras in genuine G -spectra, as introduced by Bachmann–Hoyois, is modeled 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 nonmultiplicative 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.