Diagram spaces, diagram spectra and spectra of units

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.