Semi-abelian Exact Completions

The theory of protomodular categories provides a simple and general context in which the basic theorems needed in homological algebra of groups, rings, Lie algebras and other non-abelian structures can be proved [2] [3] [4] [5] [6] [7] [9] [20]. An interesting aspect of the theory comes from the fact that there is a natural intrinsic notion of normal monomorphism [4]. Since any internal reflexive relation in a protomodular category is an equivalence relation, protomodular categories also have all the nice properties of Maltsev categories [12], so that there is in particular a good theory of centrality of equivalence relations [7] [8] [22] [25]. In many respects protomodular and, more specifically, semi-abelian categories are in the same relationship with the varieties of groups, rings and other non-abelian varieties, as abelian categories are with the varieties of abelian groups and modules over a ring. Abelian and semi-abelian categories are related by the nice “equation”