Equivariant iterated loop space theory and permutative \(G\)-categories
type: Item
instance of: scholarly article
title: Equivariant iterated loop space theory and permutative \(G\)-categories
zbMATH Open document ID: 1394.55008
published in: Algebraic \& Geometric Topology
publication date: 2017-10-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.3459
review text: It has been a classical problem solved in different ways by Boardman-Vogt, May and Segal how to recognize whether a space is an (infinite) loop space and to construct a delooping if it is. \textit{K. Shimakawa} has developed in [Publ. Res. Inst. Math. Sci. 25, No. 2, 239--262 (1989; Zbl 0677.55013)] a \(G\)-equivariant analogue of Segal's approach. The present paper considers an operadic approach to equivariant (infinite) loop space machines. The authors begin with a space-level treatment. Let us call an operad in \(G\)-spaces a \(G\)-operad. Given a \(G\)-representation \(V\), we can consider the associated little disk \(G\)-operad, where \(G\) acts on the centers of the disks. One defines an \textit{\(E_V\)-operad} to be a \(G\)-operad weakly equivalent to the little disk operad. The authors' preferred model is the Steiner operad. Given an \(E_V\)-space \(Y\), a bar construction gives a potential delooping \(\mathbb{E}_V Y\). It is shown that \(Y \to \Omega^V\mathbb{E}_V\) is a group completion if \(V\) contains the trivial representation \(\mathbb{R}^2\) and actually an equivalence if \(Y\) is \(G\)-connected. To produce \(G\)-spectra, one has to use \(G\)-\(E_\infty\)-operads, i.e.\ \(G\)-operads whose \(j\)-th space is a universal \(G\times\Sigma_j\)-space for the family of subgroups \(H\) with \(H\cap \Sigma_j = \{e\}\). Examples are the linear isometry operad and the infinite Steiner operad, both for a complete universe. If \(\mathcal{C}_G\) is a \(G\)-\(E_\infty\)-operad and \(Y\) a \(\mathcal{C}_G\)-space, the authors construct an associated orthogonal spectrum, whose zeroth space is group-completing \(Y\). An example of such a \(Y\) is the classifying space of the category of finite free \(R\)-modules for a ring \(R\) with a \(G\)-action, leading to a genuinely equivariant algebraic \(K\)-theory spectrum (as studied in detail in [\textit{M. Merling}, Math. Z. 285, No. 3--4, 1205--1248 (2017; Zbl 1365.19007)] and, using different techniques, by Barwick, Gla…
reviewed by: Lennart Meier
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 18D50
zbMATH DE Number: 6791649
zbMATH Keywords: equivariant infinite loop spaces
zbMATH Keywords: permutative categories
zbMATH Keywords: equivariant algebraic \(K\)-theory
author: Bertrand J. Guillou
author: J. P. May
MaRDI profile type: Publication
OpenAlex ID: W3105357790
arXiv ID: 1207.3459
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DOI: 10.2140/AGT.2017.17.3259
description: scientific article