Intrinsic centrality and associated classifying properties
type: Item
instance of: scholarly article
title: Intrinsic centrality and associated classifying properties
zbMATH Open document ID: 1015.18003
DOI: 10.1016/S0021-8693(02)00149-7
author: Dominique Bourn
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: Let \({\mathcal C}\) be a left exact pointed category. The category \({\mathcal C}\) is called unital provided that, for any pair of objects \((X, Y)\), the pair of morphisms \((l_X = (1_X,0) : X\to X\times Y\), \(r_Y = (0,1_Y): Y\to X\times Y\)) is jointly extremal epimorphic. It is called strongly unital provided that any relation \((r_1,r_2) : R \rightrightarrows X\) in \({\mathcal C}\) such that there exist sections \(s_1 : X\to R\), \(s_2 : X\to R\) such that \(r_1s_1 = r_2s_2 = 1_X\) and \(r_1s_2 = 0\), is necessarily the coarse relation. Let \({\mathcal C}\) be unital. A pair of morphisms \((f : X \to Z\), \(g : Y\to Z)\) is said to cooperate provided that there exists a morphism \(\varphi : X\times Y\to Z\) such that \(\varphi l_X = f\) and \(\varphi r_Y = g\). A morphism \(f : X\to Y\) is called central provided the pair \((f,1_Y)\) cooperates. It is called symmetrizable provided it has, moreover, an inverse in the monoid \(z(X,Y)\) of central morphisms from \(X\) to \(Y\). An object \(X\) is commutative (resp. abelian) provided \(1_X\) is central (resp. symmetrizable). These notions are studied in this paper and related to classical notions of additivity. These notions can be applied to any left exact category \(\mathcal E\) via the canonical fibration of points \(\Pi_E : {\mathcal P}t{\mathcal E}\to{\mathcal E}\) whose fibres are left exact pointed categories, and they are related to properties of \(\mathcal E\).
Mathematics Subject Classification ID: 18C99
Mathematics Subject Classification ID: 08B05
Mathematics Subject Classification ID: 18E05
Mathematics Subject Classification ID: 08B10
Mathematics Subject Classification ID: 18D35
zbMATH DE Number: 1868025
zbMATH Keywords: centrality
zbMATH Keywords: Mal'cev
zbMATH Keywords: unital category
zbMATH Keywords: arithmetical
zbMATH Keywords: left exact pointed category
zbMATH Keywords: additivity
reviewed by: Yves Diers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1868025