On the equivariant \(2\)-type of a \(G\)-space
type: Item
instance of: scholarly article
title: On the equivariant \(2\)-type of a \(G\)-space
zbMATH Open document ID: 0932.55011
DOI: 10.1016/S0022-4049(97)00075-3
published in: Journal of Pure and Applied Algebra
publication date: 2000-03-08T00:00:00Z
review text: A \(G\)-space \(X\) may be regarded as a diagram of its fixed point spaces \(X^H\) for \(H<G\) according to its equivariant homotopy type, since the homotopy category of \(G\)-spaces is equivalent to the homotopy category of \(O(G)^{op}\)-spaces where \(O(G)\) is the orbit category of \(G\). This paper uses the Grothendieck semidirect product construction of a small category by a functor, to define the concept of weak equivalence of small 2-categories as a functor. Finally, resting on an extension of the Grothendieck construction, the main theorem asserts that the equivariant homotopy type of a \(G\)-space \(X\) with trivial equivariant homotopy at dimensions \(g\geq 3\), is completely determined by a diagram of groupoids.
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55S45
zbMATH DE Number: 1308203
zbMATH Keywords: homotopy category
zbMATH Keywords: small category
zbMATH Keywords: equivariant homotopy type
zbMATH Keywords: Grothendieck construction
author: Manuel Bullejos
author: Julia García Cabello
author: Emilio Faro
reviewed by: Juan Antonio Pérez
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1308203