Strong cofibrations and fibrations in enriched categories
type: Item
instance of: scholarly article
title: Strong cofibrations and fibrations in enriched categories
zbMATH Open document ID: 1016.55003
DOI: 10.1007/BF02638382
published in: Archiv der Mathematik
publication date: 2003-03-27T00:00:00Z
review text: It is well known that in the category of topological spaces, if \(A\rightarrow B\) and \(B\rightarrow Y\) are cofibrations and one of them is closed, then the induced morphism \(A\times Y\cup X\times B\rightarrow X\times Y\) is a cofibration. By introducing strong cofibrations, which are equivalent to closed cofibrations in the category of topological spaces, the authors give an analogous version of this useful ``push-out product theorem''. The results are proved in the setting of enriched categories which are complete and cocomplete, tensored and cotensored. Strong cofibrations are defined by a relative homotopy lifting property. The dual concept of a strong fibration allows the authors to prove a dual result. As a consequence, structures for doing homotopical algebra are obtained. Although in important cases Quillen structures [\textit{D. G. Quillen}, Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.20903)] are not known to exist, it is possible to get fibration or cofibration structures in the sense of \textit{H.-J. Baues} [Algebraic Homotopy, Camb. Stud. Adv. Math. 15 (1989; Zbl 0688.55001)]. Finally, different examples are studied, in particular, the category of chain complexes with chain equivalences and the category of categories with equivalences of categories that are symmetric monoidal proper closed model categories.
Mathematics Subject Classification ID: 55P05
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 1889275
zbMATH Keywords: cofibrations
zbMATH Keywords: homotopy extension properties
zbMATH Keywords: stable homotopy
zbMATH Keywords: enriched categories
zbMATH Keywords: homotopical algebra
author: Roland Schwänzl
author: Rainer Vogt
reviewed by: J. Remedios
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1889275