Normal subobjects of topological groups and of topological semi-Abelian algebras
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.
scientific article; zbMATH DE number 5013544
scientific article