Diagram spaces, diagram spectra, and spectra of units

TL;DR

This paper compares different models of structured spectra and their associated infinite loop spaces, proving their equivalence and showing that two constructions of the spectrum of units of a commutative ring spectrum agree.

Contribution

It establishes Quillen equivalences between models of structured spectra and confirms the consistency of the spectrum of units construction across these models.

Findings

Models for spaces are Quillen equivalent

Infinite loop space functors agree across models

Constructions of the spectrum of units are consistent

Abstract

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra, and EKMM S-modules. Each of these categories of structured spectra has a corresponding category of structured spaces that receives the infinite loop space functor \Omega^\infty. We prove that these models for spaces are Quillen equivalent and that the infinite loop space functors \Omega^\infty agree. This comparison is then used to show that two different constructions of the spectrum of units gl_1 R of a commutative ring spectrum R agree.