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

Abstract

The category of $\mathcal{ℐ}$–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 $\mathcal{ℐ}$–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 $\Gamma$–spaces and show how the spectrum of units associated with a commutative symmetric ring spectrum arises through a chain of Quillen adjunctions.