Let be a finite group. It was observed by L.S. Scull that the original definition of the equivariant minimality in the -connected case is incorrect because of an error concerning algebraic properties. In the -disconnected case the orbit category was originally replaced by the category with one object for each component of each fixed point simplicial subsets of a -simplicial set , for all subgroups . We redefine the equivariant minimality and redevelop some results on the rational homotopy theory of disconnected -simplicial sets. To show an existence of the injective minimal model for a disconnected -simplicial set we replace by the more subtle category with one object for each 0-simplex of fixed point simplicial subsets , for all subgroups .
Si est un groupe fini, L.S. Scull a observé que la définition originale de la minimalité équivariante n’est pas correcte dans le cas -connexe par suite d’une erreur concernant des propriétés algébriques. Dans le cas -non connexe la catégorie des orbites a été remplacée par la catégorie , avec un objet pour chaque composante des sous-ensembles simpliciaux de points fixes d’un ensemble -simplicial , pour tous les sous-groupes . Nous redéfinissons la minimalité équivariante et nous redéveloppons des résultats d’homotopie rationnelle pour les ensembles -simpliciaux non connexes. Pour montrer l’existence d’un modèle minimal injectif pour un ensemble -simplicial non connexe, nous remplaçons par la catégorie plus subtile avec un objet pour chaque 0-simplexe de sous-ensembles simpliciaux de points fixes , par tous les sous- groupes .
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
Mot clés : algèbre gradué différentiel, algèbre de de Rham, $EI$-catégorie, $i$-extension élémentaire, $i$-modèle minimal, $k$-modèle compact linéairement, tour de Postnikov, quasi-isomorphisme, rationalisation, ensemble simpliciel $G$
@article{AIF_2003__53_2_625_0, author = {Golasi\'nski, Marek}, title = {On $G$-disconnected injective models}, journal = {Annales de l'Institut Fourier}, pages = {625--664}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {2}, year = {2003}, doi = {10.5802/aif.1954}, mrnumber = {1990008}, zbl = {01940706}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.1954/} }
TY - JOUR AU - Golasiński, Marek TI - On $G$-disconnected injective models JO - Annales de l'Institut Fourier PY - 2003 SP - 625 EP - 664 VL - 53 IS - 2 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.1954/ DO - 10.5802/aif.1954 LA - en ID - AIF_2003__53_2_625_0 ER -
Golasiński, Marek. On $G$-disconnected injective models. Annales de l'Institut Fourier, Volume 53 (2003) no. 2, pp. 625-664. doi : 10.5802/aif.1954. http://www.numdam.org/articles/10.5802/aif.1954/
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