Normal subobjects and abelian objects in protomodular categories
type: Item
instance of: scholarly article
title: Normal subobjects and abelian objects in protomodular categories
zbMATH Open document ID: 0969.18008
author: Dominique Bourn
published in: Journal of Algebra
publication date: 2001-10-02T00:00:00Z
review text: Let \({\mathcal E}\) be a finitely complete category. The category \({\mathcal P}t ({\mathcal E})\) of pointed objects of \({\mathcal E}\) has, as objects, splitting pairs of morphisms \((p,s)\) where \(ps\) is an identity morphism, and, as morphisms, adequate commutative diagrams. This category is fibred over \({\mathcal E}\) by the fibration \(\pi:{\mathcal P}t ({\mathcal E})\to{\mathcal E}\) assigning to \((p,s)\) the codomain of \(p\). This fibration of pointed objects of \({\mathcal E}\) is used to classify several properties of the category \({\mathcal E}\) related to internal equivalence relations, normal monomorphisms, Mal'cev operations, abelian group structures on objects and on sets of morphisms, etc. The category \({\mathcal E}\) is said to be protomodular provided that any change of base functor for \(\pi\) reflects isomorphisms. This property is a generalization of the short five lemma property. A monomorphism \(f:X\to Y\) is said to be normal to an equivalence relation \((r_1,r_2): R\rightrightarrows Y\) on \(Y\) provided that the inverse image of \(R\) by \(f\) is the coarse relation and the inverse image of \(r_1\) by \(f\) is the first projection. It means that ``\(X\) is an equivalence class of \(R\)''. This notion generalizes the notion of normal subgroups. An object \(X\) is abelian whenever it is equipped with a canonical abelian gronp structure. In a pointed protomodular category \({\mathcal E}\), this is equivalent to the existence of a canonical Mal'cev operation on \(X\) and to the fact that the diagonal of \(X\) is normal. The category \({\mathcal G}r\) of groups is a good example of such a category in which all the results of this paper apply, but other examples are given.
Mathematics Subject Classification ID: 18D30
Mathematics Subject Classification ID: 18A20
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D35
zbMATH DE Number: 1475242
zbMATH Keywords: Mal'cev category
zbMATH Keywords: abelian object
zbMATH Keywords: normal monomorphisms
zbMATH Keywords: protomodular category
reviewed by: Yves Diers
MaRDI profile type: Publication
OpenAlex ID: W2084973715
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DOI: 10.1006/JABR.1999.8249
description: scientific article; zbMATH DE number 1475242