Normal subobjects and abelian objects in protomodular categories (Q1570871)

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.