A van Kampen theorem for equivariant fundamental groupoids
type: Item
instance of: scholarly article
title: A van Kampen theorem for equivariant fundamental groupoids
zbMATH Open document ID: 1154.55007
author: Manuel Bullejos
author: Laura Scull
published in: Journal of Pure and Applied Algebra
publication date: 2008-06-26T00:00:00Z
review text: The equivariant fundamental groupoid of a \(G\)-space \(X\) is a category which generalizes the usual fundamental groupoid. Its construction due to tom Dieck can be presented using the Grothendieck semidirect product construction which gives a single category \(\int_{\mathcal C}F\) from a functor \(F : {\mathcal C}^{op} \to Cat\). The objects of \(\int_{\mathcal C}F\) are pairs \((C,X)\) with \(C\) an object in \({\mathcal C}\) and \(X\) an object in \(F(C)\), and the arrows \((C,X) \to (C',Y)\) are pairs \((f,v)\) with \(f : C \to C'\) an arrow in \({\mathcal C}\) and \(v: X \to F(f)(Y)\) an arrow in \(F(C)\). Denote by \({\mathcal O}_G\) the usual orbit category and consider the functor \(F\) from \({\mathcal O}_G^{op}\) to the category of groupoids, \(Gpd\), which associates to \(G/H\) the fundamental group of \(X^H\). The equivariant fundamental groupoid is the category obtained by applying the Grothendieck semidirect product to the functor \(F\). Note that this category is a category fibered in groupoids over \({\mathcal O}_G\). The main result of the paper is a generalization of Van Kampen's theorem and says that, if \(U_1\) and \(U_2\) are open \(G\)-subsets covering \(X\), then the equivariant fundamental groupoid of \(X\) is the pushout (in Cat and in Fib\(({\mathcal O}_G)\)) of the equivariant fundamental groupoids associated to \(U_1\) and \(U_2\).
reviewed by: Yves Félix
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 5293353
zbMATH Keywords: fundamental group
zbMATH Keywords: van Kampen theorem
zbMATH Keywords: groupoid
MaRDI profile type: Publication
full work available at URL: j.jpaa.2007.12.005
OpenAlex ID: W2085407591
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DOI: 10.1016/J.JPAA.2007.12.005
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description: scientific article; zbMATH DE number 5293353