Volume 13, issue 4 (2013)

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Diagram spaces, diagram spectra and spectra of units

John A Lind

Algebraic & Geometric Topology 13 (2013) 1857–1935
Abstract

This article compares the infinite loop spaces associated to symmetric spectra, orthogonal spectra and EKMM $S\phantom{\rule{0.3em}{0ex}}$–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.

Keywords
$E_\infty$–spaces, infinite loop spaces, structured ring spectra, symmetric spectra, orthogonal spectra, EKMM, spectra of units
Mathematical Subject Classification 2010
Primary: 55P42, 55P43, 55P47, 55U35, 55U40
Secondary: 55P48, 18G55