On G-disconnected injective models
Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 625-664.

Let G be a finite group. It was observed by L.S. Scull that the original definition of the equivariant minimality in the G-connected case is incorrect because of an error concerning algebraic properties. In the G-disconnected case the orbit category 𝒪(G) was originally replaced by the category 𝒪(G,X) with one object for each component of each fixed point simplicial subsets X H of a G-simplicial set X, for all subgroups HG. We redefine the equivariant minimality and redevelop some results on the rational homotopy theory of disconnected G-simplicial sets. To show an existence of the injective minimal model X for a disconnected G-simplicial set X we replace 𝒪(G,X) by the more subtle category 𝒪 ˜(G,X) with one object for each 0-simplex of fixed point simplicial subsets X H , for all subgroups HG.

Si G est un groupe fini, L.S. Scull a observé que la définition originale de la minimalité équivariante n’est pas correcte dans le cas G-connexe par suite d’une erreur concernant des propriétés algébriques. Dans le cas G-non connexe la catégorie des orbites 𝒪(G) a été remplacée par la catégorie 𝒪(G,X), avec un objet pour chaque composante des sous-ensembles simpliciaux de points fixes X H d’un ensemble G-simplicial X, pour tous les sous-groupes HG. Nous redéfinissons la minimalité équivariante et nous redéveloppons des résultats d’homotopie rationnelle pour les ensembles G-simpliciaux non connexes. Pour montrer l’existence d’un modèle minimal injectif X pour un ensemble G-simplicial X non connexe, nous remplaçons 𝒪(G,X) par la catégorie plus subtile 𝒪 ˜(G,X) avec un objet pour chaque 0-simplexe de sous-ensembles simpliciaux de points fixes X H , par tous les sous- groupes HG.

DOI: 10.5802/aif.1954
Classification: 55P62,  55P91,  16W80,  18G30
Keywords: differential graded algebra, de Rham algebra, EI-category, i-elementary extension, i-minimal model, linearly compact (complete) k-module, Postnikov tower, quasi-isomorphism, rationalization, G-simplicial set
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Golasiński, Marek. On $$-disconnected injective models. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 625-664. doi : 10.5802/aif.1954. https://aif.centre-mersenne.org/articles/10.5802/aif.1954/

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