The Hurewicz model structure on simplicial \(R\)-modules
type: Item
instance of: scholarly article
title: The Hurewicz model structure on simplicial \(R\)-modules
author: Arnaud Ngopnang Ngompé
published in: Journal of Homotopy and Related Structures
publication date: 2024-11-14T00:00:00Z
review text: For a nice bicomplete abelian category \(\mathcal{A}\), two of the well-known standard model structures on the category \(\mbox{Ch}(\mathcal{A})\) of unbounded chain complexes of objects of \(\mathcal{A}\) are the projective model structure, for which weak equivalences are quasi-isomorphisms and fibrations are degreewise epimorphisms, and the Hurewicz model structure, for which weak equivalences are chain homotopy equivalences and fibrations are degreewise split epimorphisms. \N\NThe Hurewicz model structure on \(\mbox{Ch}(\mathcal{A})\) was introduced by \textit{M. Golasiński} and \textit{G. Gromadzki} [Colloq. Math. 47, 173--178 (1982; Zbl 0532.55025)]. It was later also constructed using different methods by e.g., [\textit{J. D. Christensen} and \textit{M. Hovey}, Math. Proc. Camb. Philos. Soc. 133, No. 2, 261--293 (2002; Zbl 1016.18008)]. \textit{D. G. Quillen} [Homotopical algebra. Berlin-Heidelberg-New York: Springer-Verlag (1967; Zbl 0168.20903)] introduced the projective model structure on the category \(\mbox{Ch}_{\ge 0}(\mathcal{A})\) of non-negatively graded chain complexes of objects of \(\mathcal{A}\), i.e., with quasi-isomorphisms as weak equivalences and degreewise monomorphisms with degreewise projective cokernel as cofibrations. \N\NThe Dold-Kan correspondence induces model structures on the category of simplicial \(R\)-modules \(s\mbox{Mod}_R\). The author presents a description of the two induced model categories on \(s\mbox{Mod}_R\) and some of their properties, notably the fact that both are monoidal. Furthermore, fibrations and cofibrations are characterized in terms of homotopy lifting and homotopy extension, respectively.
reviewed by: Marek Golasiński
Mathematics Subject Classification ID: 18G31
Mathematics Subject Classification ID: 18N40
Mathematics Subject Classification ID: 55U10
zbMATH DE Number: 7944385
zbMATH Keywords: chain complex
zbMATH Keywords: enriched model category
zbMATH Keywords: Dold-Kan correspondence
zbMATH Keywords: homotopy equivalence
zbMATH Keywords: model category
zbMATH Keywords: monoidal
MaRDI profile type: Publication
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DOI: 10.1007/S40062-024-00359-0
description: scientific article; zbMATH DE number 7944385