# 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

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topGolasiński, 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 -

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