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Marek Golasiński
On $G$-disconnected injective models
(Sur les $G$-modèles injectifs non connexess)
Annales de l'institut Fourier, 53 no. 2 (2003), p. 625-664, doi: 10.5802/aif.1954
Article PDF | Reviews MR 1990008 | Zbl 01940706
Class. Math.: 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

Résumé - Abstract

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

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