Classification of dicoverings
type: Item
instance of: scholarly article
title: Classification of dicoverings
zbMATH Open document ID: 1200.55016
author: Lisbeth Fajstrup
published in: Topology and its Applications
publication date: 2010-09-24T00:00:00Z
full work available at URL: R-2009-13.pdf
review text: In a previous paper [Homology Homotopy Appl. 5, No.~2, 1--17, electronic only (2003; Zbl 1030.68057)], the author proposed a construction of the universal dicovering \(\hat{X}\) of a well-pointed \(d\)-space \((X,x)\) (i.e. in which all points are the target of a \(d\)-path with source \(x\)). She provided examples where the universal map failed to be continuous and it was unclear whether the solution was to restrict the spaces \(X\) or the dicoverings allowed, or to change the topology on \(\hat{X}\). The author proves in this paper that the latter was the good one. The good space is obtained by taking the boxification of \(\hat{X}\) (the effect is to add opens). The boxification is the coreflector associated with the full coreflective subcategory of \(d\)-spaces generated by directed cubes. It is also proved in this paper that this subcategory is actually generated by the asynchronous segment \([0,1]\) and the directed segment. The author shows that the same proof applies for \(\Delta\)-generated spaces which are therefore generated by the segment \([0,1]\). In the last part of the paper, the dicoverings of a well pointed \(d\)-space are classified as quotients of the universal dicovering under congruence relations.
reviewed by: Philippe Gaucher
Mathematics Subject Classification ID: 55P99
zbMATH DE Number: 5791181
zbMATH Keywords: directed homotopy
zbMATH Keywords: coreflective subcategory
zbMATH Keywords: universal dicovering
MaRDI profile type: Publication
OpenAlex ID: W2026579352
cites work: Fibrations of graphs
cites work: Reparametrizations of continuous paths
cites work: Dicovering spaces
cites work: Covering space theory for directed topology
cites work: Q4453908
cites work: A convenient category of locally preordered spaces
cites work: A convenient category for directed homotopy
cites work: Erratum to ``On the expressiveness of higher dimensional automata''
DOI: 10.1016/J.TOPOL.2010.07.020
Recommended article: Saturating directed spaces
Recommended article: Q3580752
Recommended article: Pair component categories for directed spaces
Recommended article: Models and van Kampen theorems for directed homotopy theory
Recommended article: Six model categories for directed homotopy
Recommended article: Simplicial models for trace spaces. II: General higher dimensional automata
Recommended article: Q4837714
Recommended article: Dicovering spaces
Recommended article: Q3461029
description: scientific article; zbMATH DE number 5791181