Some remarks on Cartan-Eilenberg categories
type: Item
instance of: scholarly article
title: Some remarks on Cartan-Eilenberg categories
zbMATH Open document ID: 1266.18017
published in: Collectanea Mathematica
publication date: 2012-12-04T00:00:00Z
review text: Cartan-Eilenberg categories offer an alternative approach to homotopical algebra. They were introduced in [\textit{F. Guillén}, J. Pure Appl. Algebra 214, No. 2, 140--164 (2010; Zbl 1198.18006)]. The basic ingredients of the notion are a category, \(\mathcal{C}\), with two classes, \(\mathcal{S}\subseteq \mathcal{W}\), of distinguished morphism called \textit{strong} and \textit{weak equivalences} such that, for each object, \(X\), of \(\mathcal{C}\), there is a `cofibrant' object, \(M\), and a morphism \(q: M\to X\) in \(\mathcal{C}[\mathcal{S}^{-1}]\), which is an isomorphism in \(\mathcal{C}[\mathcal{W}^{-1}]\). (`Cofibrant' is defined by a simple lifting property.) This allows for a useful notion of minimal object analogous to that studied in Sullivan's theory of minimal models. In this paper, which can be considered as an appendix to the one cited above, two non-examples are given, then various general classes of example are studied with application given to finite topological spaces, deformation theory and chain homology functors.
reviewed by: Timothy Porter
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 6112452
author: Pere Pascual-Gainza
MaRDI profile type: Publication
full work available at URL: s13348-011-0037-9
OpenAlex ID: W2127366531
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DOI: 10.1007/S13348-011-0037-9
description: scientific article; zbMATH DE number 6112452