Intrinsic centrality and associated classifying properties (Q1858212)

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 \(\mathcal E\).