On G -disconnected injective models

Marek Golasiński[1]

  • [1] Nicholas Copernicus University, Faculty of Mathematics and Computer Science, Chopina 12/18, 87-100 Toruń (Pologne)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 2, page 625-664
  • ISSN: 0373-0956

Abstract

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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 H ⊆ 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 𝒪 ˜ ( G , X ) with one object for each 0-simplex of fixed point simplicial subsets X H , for all subgroups H ⊆ G .

How to cite

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Golasiński, Marek. "On $G$-disconnected injective models." Annales de l’institut Fourier 53.2 (2003): 625-664. <http://eudml.org/doc/116047>.

@article{Golasiński2003,
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 $\{\mathcal \{O\}\}(G)$ was originally replaced by the category $\{\mathcal \{O\}\}(G,X)$ 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 $\{\mathcal \{M\}\}_X$ for a disconnected $G$-simplicial set $X$ we replace $\{\mathcal \{O\}\}(G,X)$ by the more subtle category $\tilde\{\mathcal \{O\}\}(G,X)$ with one object for each 0-simplex of fixed point simplicial subsets $X^H$, for all subgroups $H\subseteq G$.},
affiliation = {Nicholas Copernicus University, Faculty of Mathematics and Computer Science, Chopina 12/18, 87-100 Toruń (Pologne)},
author = {Golasiński, Marek},
journal = {Annales de l’institut Fourier},
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; EI-category; i-elementary extension; i-minimal model; linearly compact (complete) k-module; G-simplicial set},
language = {eng},
number = {2},
pages = {625-664},
publisher = {Association des Annales de l'Institut Fourier},
title = {On $G$-disconnected injective models},
url = {http://eudml.org/doc/116047},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Golasiński, Marek
TI - On $G$-disconnected injective models
JO - Annales de l’institut Fourier
PY - 2003
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 2
SP - 625
EP - 664
AB - 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 ${\mathcal {O}}(G)$ was originally replaced by the category ${\mathcal {O}}(G,X)$ 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 ${\mathcal {M}}_X$ for a disconnected $G$-simplicial set $X$ we replace ${\mathcal {O}}(G,X)$ by the more subtle category $\tilde{\mathcal {O}}(G,X)$ with one object for each 0-simplex of fixed point simplicial subsets $X^H$, for all subgroups $H\subseteq G$.
LA - eng
KW - 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; EI-category; i-elementary extension; i-minimal model; linearly compact (complete) k-module; G-simplicial set
UR - http://eudml.org/doc/116047
ER -

References

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