Homotopy theory of cofibration categories
type: Item
instance of: scholarly article
title: Homotopy theory of cofibration categories
zbMATH Open document ID: 1364.55018
author: Karol Szumiło
published in: Homology, Homotopy and Applications
publication date: 2017-01-20T00:00:00Z
review text: The author constructs a fibration category theory for the category of cofibration categories. Let \(\mathsf{CofCat}\) be the category of small cofibration categories, with morphisms given by exact functors between them. The definition of cofibration category structure adopted here is almost the dual version of Brown's original one [\textit{K. S. Brown}, Trans. Am. Math. Soc. 186, 419--458 (1973; Zbl 0245.55007)]. An exact functor of cofibration categories \(F:\mathcal{C}\rightarrow\mathcal{D}\) is defined to be a \textit{weak equivalence} if it induces an equivalence between the homotopy categories \(\mathrm{Ho}\mathcal{C}\rightarrow\mathrm{Ho}\mathcal{D}\). In order to define a \textit{fibration}, some properties are required from an exact functor. Those properties allow factorizations and the existence of pullbacks along cofibrations in \(\mathsf{CofCat}\). In addition, a technical property is also required in order to allow that acyclic fibrations (exact functors which are both a weak equivalence and a fibration) are stable under pullback. Then, the main theorem shows that the category \(\mathsf{CofCat}\) equipped with \textit{weak equivalences} and \textit{fibrations} has a complete fibration category structure.
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 6676577
zbMATH Keywords: fibration category
zbMATH Keywords: cofibration category
zbMATH Keywords: exact functor
reviewed by: J. Remedios
MaRDI profile type: Publication
full work available at URL: hha.2016.v18.n2.a19
OpenAlex ID: W2554367662
DOI: 10.4310/HHA.2016.V18.N2.A19
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description: scientific article; zbMATH DE number 6676577