Homotopy classification of braided graded categorical groups
type: Item
instance of: scholarly article
title: Homotopy classification of braided graded categorical groups
zbMATH Open document ID: 1139.55002
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-26T00:00:00Z
review text: Let \(G\) be a group, \(M\) and \(N\) be \(G\)-modules, \(H^n_{G,ab}(M,-)\) be the Quillen cohomology groups of the pointed \(G\)-space \(K(M, 2)\) in the homotopy category of \(G\)-spaces, \(Ho\mathbb{S}^G_*\), with respect to the closed model structure where weak equivalences are those \(G\)-equivariant pointed maps that are weak equivalences on the underlying spaces, that is, \[ H^n_{G,ab}(M, N)= \text{Hom}_{Ho\mathbb{S}^G_*}(K(M, 2), K(N, n+1)). \] This paper shows that equivariant weak homotopy types of pointed \(G\)-spaces \(X\) with only two non-trivial homotopy groups at dimensions 2 and \(n\) are classified by the elements of the cohomology groups \(H^n_{G,ab}(\pi_2 X,\pi_n X)\) and then, the authors get their main result as follows: The homotopy category of \(G\)-equivariant pointed simply-connected 3-types is equivalent to the homotopy category of braided \(G\)-graded categorical groups.
Mathematics Subject Classification ID: 55N25
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 5129950
zbMATH Keywords: Quillen cohomology groups
zbMATH Keywords: equivariant weak homotopy types
zbMATH Keywords: 3-types
author: Antonio M. Cegarra
author: E. Khmaladze
reviewed by: Wenhuai Shen
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.06.011
OpenAlex ID: W2114282879
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DOI: 10.1016/J.JPAA.2006.06.011
description: scientific article; zbMATH DE number 5129950