Strongly fibered objects and spaces
type: Item
instance of: scholarly article
title: Strongly fibered objects and spaces
zbMATH Open document ID: 1405.55022
author: Luciano Stramaccia
published in: Topology and its Applications
publication date: 2018-01-05T00:00:00Z
review text: Let \(\mathcal{C}\) be a groupoid enriched (g.e.) category with a model category structure having a functorial cylinder, then it is known that the category, \(Pro(\mathcal{C})\), of inverse systems in \(\mathcal{C}\) also has a g.e. model category structure. This allows discussions of strong shape theory within the context of \(\mathcal{C}\) and its model category, relative to a full subcategory, \(\mathcal{K}\), all of whose objects are fibrant; see [\textit{L. Stramaccia}, J. Homotopy Relat. Struct. 12, No. 2, 433--446 (2017; Zbl 1372.55015)]. (The classical case was \(\mathcal{C}=\mathsf{TOP}\) and \(\mathcal{K}=\mathsf{ANR}\).) Generalising a classical topological notion, the author introduces the notion of a strongly fibered object, corresponding to limits of fibrant inverse systems in \(Pro(\mathcal{C})\), and denotes by \(\mathcal{F(K)}\) the full subcategory of \(\mathcal{C}\) determined by these objects. The main theorem shows that the shape theory of the pair \((\mathcal{C},\mathcal{F(K)})\) realises the strong shape category of \((\mathcal{C},\mathcal{K})\), thus generalising the classical result in the topological context.
reviewed by: Timothy Porter
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55P55
Mathematics Subject Classification ID: 18D20
zbMATH DE Number: 6822542
zbMATH Keywords: inverse system
zbMATH Keywords: groupoid enriched category
zbMATH Keywords: pseudo-natural transformation
zbMATH Keywords: homotopy limit
zbMATH Keywords: shape category
zbMATH Keywords: strong shape category
MaRDI profile type: Publication
full work available at URL: j.topol.2017.11.003
OpenAlex ID: W2768443032
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DOI: 10.1016/J.TOPOL.2017.11.003
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description: scientific article; zbMATH DE number 6822542