Stable homotopy monomorphisms
type: Item
instance of: scholarly article
title: Stable homotopy monomorphisms
zbMATH Open document ID: 0873.18001
DOI: 10.1023/A:1008699129699
author: R. W. Kieboom
published in: Applied Categorical Structures
publication date: 1997-10-27T00:00:00Z
review text: The concept of monomorphism in a category can be defined using an appropriate pullback diagram. But in the homotopy category of \({\mathcal T} {\mathcal O} {\mathcal P}\) pullbacks do not generally exist. Then \textit{M. Mather} [Can. J. Math. 28, 225-263 (1976; Zbl 0351.55005)] introduced another notion of homotopy pullbacks which do exist. Now the author of this paper modifies the notion of homotopy monomorphism obtained by applying the pullback characterization using Mather's homotopy pullback. The author shows that these modified homotopy monomorphisms are exactly those homotopy monomorphisms (in the usual sense) which are homotopy pullback stable, hence the terminology ``stable'' homotopy monomorphism. Also the author links these stable homotopy monomorphism to monomorphisms and products in the track homotopy category over a fixed space. In the second part of the paper the author studies whether a (weak) fibration is also a stable homotopy monomorphism. It is shown that the class of (weak) fibrations with this additional property coincides with the class of ``double'' (weak) fibrations. (The double (weak) covering homotopy property introduced in this paper is a stronger version of the \((W)CHP\) in which the final maps of the homotopies involved play the same role as the initial maps).
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 55R65
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 1036411
zbMATH Keywords: weak fibrations
zbMATH Keywords: homotopy pullback stable
zbMATH Keywords: stable homotopy monomorphism
zbMATH Keywords: track homotopy category
reviewed by: Ioan Pop
MaRDI profile type: Publication
full work available at URL: a:1008699129699
OpenAlex ID: W2012197063
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description: scientific article; zbMATH DE number 1036411