Group completion and units in I-spaces

TL;DR

This paper studies I-spaces, a category modeling E-infinity spaces, and simplifies the understanding of their group completion and units, connecting to Gamma-spaces and units spectra via Quillen adjunctions.

Contribution

It provides a simplified framework for group completion and units in E-infinity spaces using I-spaces, clarifying their relation to Gamma-spaces and units spectra.

Findings

Simplifies the theory of group completion in I-spaces.

Clarifies the relation between units in E-infinity spaces and Gamma-spaces.

Shows how units spectra arise through Quillen adjunctions.

Abstract

The category of 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-infinity spaces. Working in the category of I-spaces enables us to simplify and strengthen previous work on group completion and units of E-infinity 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.