Many homotopy categories are homotopy categories
type: Item
instance of: scholarly article
title: Many homotopy categories are homotopy categories
zbMATH Open document ID: 1097.55013
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: A well known theorem of \textit{A. Strøm} [Arch. Math. 23, 435--441 (1972; Zbl 0261.18015)] states that the category of topological spaces admits a model category structure where weak equivalences are homotopy equivalences and fibrations are Hurewicz fibrations. The paper generalizes this result as follows. Let \(\mathcal A\) be a complete and cocomplete category which is enriched, tensored and cotensored over the category of compactly generated topological spaces. For objects \(X, Y\in\mathcal A\) one can define a homotopy of morphisms from \(X\) to \(Y\) as a map \(X\otimes I \rightarrow Y\) (where \(I\) is the unit interval). This in turn can be used to define the notions of a homotopy equivalence and Hurewicz fibration in \(\mathcal A\). The author shows that these classes of morphisms define a model category structure on \(\mathcal A\) provided that an additional hypothesis concerning behavior of certain colimits is satisfied. Beside the category of topological spaces this result applies to the categories of \(G\)-spaces, prespectra, spectra, \(S\)-modules and others.
reviewed by: Bernard Badzioch
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 5013525
zbMATH Keywords: model category
zbMATH Keywords: homotopy category
zbMATH Keywords: Hurewicz fibration
author: Michael Cole
MaRDI profile type: Publication
full work available at URL: j.topol.2005.02.006
OpenAlex ID: W2012099811
cites work: Q4896538
cites work: Q4716283
cites work: Q4220599
cites work: Equivariant stable homotopy theory. With contributions by J. E. McClure
cites work: Homotopical algebra
cites work: Strong cofibrations and fibrations in enriched categories
cites work: The homotopy category is a homotopy category
DOI: 10.1016/J.TOPOL.2005.02.006
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description: scientific article; zbMATH DE number 5013525