On $G$-disconnected injective models

Golasiński, Marek

Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 625-664.

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 $𝒪\left(G\right)$ 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 $H\subseteq G$. 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 ${ℳ}_X$ pour un ensemble $G$-simplicial $X$ non connexe, nous remplaçons $𝒪(G,X)$ par la catégorie plus subtile $\widetilde{𝒪}(G,X)$ avec un objet pour chaque 0-simplexe de sous-ensembles simpliciaux de points fixes $X^H$, par tous les sous- groupes $H\subseteq G$.

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 $𝒪\left(G\right)$ 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 $H\subseteq G$. 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 ${ℳ}_X$ for a disconnected $G$-simplicial set $X$ we replace $𝒪(G,X)$ by the more subtle category $\widetilde{𝒪}(G,X)$ with one object for each 0-simplex of fixed point simplicial subsets $X^H$, for all subgroups $H\subseteq G$.

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