Units of equivariant ring spectra

TL;DR

This paper establishes a Quillen equivalence between equivariant $ ext{Gamma}$-spaces and equivariant $ ext{E}_ ext{infty}$-spaces, and defines units of equivariant ring spectra, linking algebraic structures to connective equivariant spectra.

Contribution

It proves a Quillen equivalence between models of equivariant spectra and defines units of equivariant ring spectra using equivariant $ ext{Gamma}$-spaces.

Findings

Equivariant $ ext{Gamma}$-spaces and $ ext{E}_ ext{infty}$-spaces are Quillen equivalent.

Units of equivariant ring spectra determine connective equivariant spectra.

The paper provides model category structures for these equivalences.

Abstract

It is well known that very special $\Gamma$-spaces and grouplike $\E_\infty$ spaces both model connective spectra. Both these models have equivariant analogues. Shimakawa defined the category of equivariant $\Gamma$-spaces and showed that special equivariant $\Gamma$-spaces determine positive equivariant spectra. Costenoble and Waner showed that grouplike equivariant $\E_\infty$-spaces determine connective equivariant spectra. We show that with suitable model category structures the category of equivariant $\Gamma$-spaces is Quillen equivalent to the category of equivariant $\E_\infty$ spaces. We define the units of equivariant ring spectra in terms of equivariant $\Gamma$-spaces and show that the units of an equivariant ring spectrum determines a connective equivariant spectrum.