G-symmetric spectra, semistability and the multiplicative norm

TL;DR

This paper develops a homotopy theory framework for G-symmetric spectra, providing a model for equivariant stable homotopy that bridges existing models and explores semistability and multiplicative structures.

Contribution

It introduces a new model for G-symmetric spectra that interpolates between existing models, and analyzes semistability and multiplicative norm properties within this framework.

Findings

The model is Quillen equivalent to existing equivariant spectra models.

Constructs model structures on module, algebra, and commutative algebra categories.

Describes homotopical properties of the multiplicative norm.

Abstract

In this paper we develop the basic homotopy theory of G-symmetric spectra (that is, symmetric spectra with a G-action) for a finite group G, as a model for equivariant stable homotopy with respect to a G-set universe. This model lies in between Mandell's equivariant symmetric spectra and the G-orthogonal spectra of Mandell and May and is Quillen equivalent to the two. We further discuss equivariant semistability, construct model structures on module, algebra and commutative algebra categories and describe the homotopical properties of the multiplicative norm in this context.