Injective models of \(G\)-disconnected simplicial sets
type: Item
instance of: scholarly article
title: Injective models of \(G\)-disconnected simplicial sets
zbMATH Open document ID: 0886.55012
publication date: 1997-12-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75271
review text: The author extends the rational homotopy theory on \(G\)-disconnected simplicial sets being not necessary of finite \(G\)-type. Instead of the orbit category \(O(G)\), in this paper the author works over the category \(O(G,X)\) with one object for each component of each fixed point simplicial subset \(X^H\) of a \(G\)-simplicial set \(X\) for all subgroups \(H\) in \(G\). First he investigates the category \(kI\)-Mod of covariant functors on a small category \(I\) to the category of \(k\)-modules over a field \(k\) and presents some considerations about injective objects in the category \(kI\)-Mod. Then he extends these on the category \(I\)-\(DGA_k\) of functors from an \(EI\)-category \(I\) (all endomorphisms are isomorphisms) to the category \(DGA_k\). He presents the existence of an injective minimal model for a complete injective \(kI\)-algebra \(A\) for an \(EI\)-category \(I\). He shows that on de Rham algebra \(A^*_X\) of rational polynomial forms on a simplicial set \(X\) there is a natural complete linear topology. Finally, for a \(G\)-simplicial set \(X\) he associates the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\) \((\mathbb{Q}\) the field of rationals) and shows the existence of an injective minimal model for the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\). The main result describes the rational homotopy type of a nilpotent \(G\)-simplicial set \(X\) by means of injective minimal model of the de Rham \(\mathbb{Q} O(G,X)\)-algebra \(A^*_X\).
Mathematics Subject Classification ID: 55P62
Mathematics Subject Classification ID: 55P91
Mathematics Subject Classification ID: 16W80
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 1092702
zbMATH Keywords: differential graded algebra
zbMATH Keywords: de Rham algebra
zbMATH Keywords: \(EI\)-category
zbMATH Keywords: Grothendieck construction
zbMATH Keywords: \(i\)-minimal model
zbMATH Keywords: linearly compact (complete)
zbMATH Keywords: \(k\)-module
zbMATH Keywords: Postnikov tower
zbMATH Keywords: simplicial set
zbMATH Keywords: injective model
zbMATH Keywords: equivariant homotopy theory
zbMATH Keywords: simplicial object
author: Marek Golasiński
reviewed by: Corina Mohorianu
MaRDI profile type: Publication
Wikidata QID: Q114013479
OpenAlex ID: W2047488166
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DOI: 10.5802/AIF.1607
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1092702