Equivariant $\Gamma$-spaces

TL;DR

This paper develops a homotopy theory framework for $ ext{Gamma}$-$G$-spaces, establishing model structures, adjunctions, and equivalences with $G$-symmetric spectra, extending non-equivariant results to the equivariant setting.

Contribution

It introduces model structures and Quillen adjunctions for $ ext{Gamma}$-$G$-spaces, and proves they model connective equivariant stable homotopy theory, extending classical non-equivariant results.

Findings

Established level and stable model structures on $ ext{Gamma}$-$G$-spaces.

Proved $ ext{Gamma}$-$G$-spaces model connective equivariant stable homotopy.

Showed the functor to $G$-symmetric spectra commutes with the derived smash product.

Abstract

The aim of this note is to provide a comprehensive treatment of the homotopy theory of $\Gamma$-$G$-spaces for $G$ a finite group. We introduce two level and stable model structures on $\Gamma$-$G$-spaces and exhibit Quillen adjunctions to $G$-symmetric spectra with respect to a flat level and a stable flat model structure respectively. Then we give a proof that $\Gamma$-$G$-spaces model connective equivariant stable homotopy theory along the lines of the proof in the non-equivariant setting given by Bousfield and Friedlander. Furthermore, we study the smash product of $\Gamma$-$G$-spaces and show that the functor from $\Gamma$-$G$-spaces to $G$-symmetric spectra commutes with the derived smash product. Finally, we show that there is a good notion of geometric fixed points for $\Gamma$-$G$-spaces.