Normal subobjects of topological groups and of topological semi-Abelian algebras
type: Item
instance of: scholarly article
title: Normal subobjects of topological groups and of topological semi-Abelian algebras
zbMATH Open document ID: 1089.18005
author: Dominique Bourn
published in: Topology and its Applications
publication date: 2006-03-20T00:00:00Z
review text: This paper investigates properties of the category \({\mathcal G} r({\mathcal T} op)\) of topological groups. On the one hand, it is a category which is topological over the category \({\mathcal G} r\) of groups, but on the other hand, being the category of internal groups in the category \({\mathcal T}op\) of topological spaces, it is also a so-called protomodular category. This notion of protomodularity previously defined by the author means that it is a finitely complete pointed category in which any split epimorphism \(f:X \to Y\) with section \(s:Y\to X\) gives rise to a supremum \((\text{ker}(f), s)\) of subobjects of \(X\). Moreover being a Barr regular category, it is a so-called homological category, i.e., a category in which all the homological lemmas do hold. The author investigates the normal subobjects, the commutator theory and exact sequences in this category. The results are extended to categories of topological semi-abelian algebras.
Mathematics Subject Classification ID: 18D35
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 22A99
Mathematics Subject Classification ID: 13J99
Mathematics Subject Classification ID: 18B30
zbMATH DE Number: 5013544
zbMATH Keywords: protomodular category
zbMATH Keywords: commutator
reviewed by: Yves Diers
MaRDI profile type: Publication
full work available at URL: j.topol.2005.04.005
OpenAlex ID: W1981747303
cites work: Q4693062
cites work: Q5628320
cites work: Q4827951
cites work: Topological semi-abelian algebras
cites work: Q3285686
cites work: Q3801758
cites work: Q3978968
cites work: Normal subobjects and abelian objects in protomodular categories
cites work: \(3\times 3\) lemma and protomodularity
cites work: Q4663626
cites work: Q3156493
cites work: Centrality and normality in protomodular categories
cites work: Q4707050
cites work: Q4776083
cites work: Groups with Multiple Operators
cites work: Semi-abelian categories
cites work: On the categories of general topology and topological algebra
DOI: 10.1016/J.TOPOL.2005.04.005
description: scientific article; zbMATH DE number 5013544