A van Kampen theorem for equivariant fundamental groupoids
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 \(U_2\).
scientific article; zbMATH DE number 5293353
scientific article