#### Volume 13, issue 2 (2013)

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Group completion and units in $\mathcal{I}\mkern -1mu$–spaces

### Steffen Sagave and Christian Schlichtkrull

Algebraic & Geometric Topology 13 (2013) 625–686
##### Abstract

The category of $\mathsc{ℐ}$–spaces is the diagram category of spaces indexed by finite sets and injections. This is a symmetric monoidal category whose commutative monoids model all ${E}_{\infty }$–spaces. Working in the category of $\mathsc{ℐ}$–spaces enables us to simplify and strengthen previous work on group completion and units of ${E}_{\infty }$–spaces. As an application we clarify the relation to $\Gamma \phantom{\rule{0.3em}{0ex}}$–spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions.

##### Keywords
$E_{\infty}$–spaces, group completion, units of ring spectra, $\Gamma$–spaces
Primary: 55P48
Secondary: 55P43