G-symmetric monoidal categories of modules over equivariant commutative ring spectra

Abstract

We describe the multiplicative structures that arise on categories of equivariant modules over certain equivariant commutative ring spectra. Building on our previous work on N-infinity ring spectra, we construct categories of equivariant operadic modules over N-infinity rings that are structured by equivariant linear isometries operads. These categories of modules are endowed with equivariant symmetric monoidal structures, which amounts to the structure of an "incomplete Mackey functor in homotopical categories". In particular, we construct internal norms which satisfy the double coset formula. We regard the work of this paper as a first step towards equivariant derived algebraic geometry.

Revised to include appendix on compact Lie groups, reflect referee comments

References in corpus (8)

• Global homotopy theory
• The spectrum of the equivariant stable homotopy category of a finite group
• The $C_2$-spectrum $Tm{f}_1(3)$ and its invertible modules
• Equivariant symmetric monoidal structures
• Incomplete Tambara functors
• Parametrized higher category theory and higher algebra: A general introduction
• Constructing equivariant spectra via categorical Mackey functors
• Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

Cited by in corpus (10)

• Equivariant symmetric monoidal structures
• Parametrized higher category theory and higher algebra: A general introduction
• Invertible $K(2)$-Local $E$-Modules in $C_4$-Spectra
• Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum
• Equivariant Eilenberg-Mac Lane spectra in cyclic $p$-groups
• Bousfield Localization and Eilenberg-Moore Categories
• The genuine operadic nerve
• Idempotent characters and equivariantly multiplicative splittings of K-theory
• Goerss--Hopkins obstruction theory for $\infty$-categories
• ${\mathbb{E}}_{\infty }$ automorphisms of motivic Morava $E$-theories