Abelian Groupoids and Non-pointed Additive Categories

We show that, in any Mal’tsev (and a fortiori protomodular) category E, not only the fibre GrdXE of internal groupoids above the object X is a naturally Mal’tsev category, but moreover it shares with the category Ab of abelian groups the property following which the domain of any split epimorphism is isomorphic with the direct sum of its codomain with its kernel. This allows us to point at a new class of “non-pointed additive” categories which is necessarily protomodular. Actually this even gives rise to a larger classification table of non-pointed additive categories which gradually take place between the class of naturally Mal’tsev categories [16] and the one of essentially affine categories [5]. As an application, when furthermore the ground category E is efficiently regular, we get a new way to produce Bear sums in the fibres GrdXE and, more generally, in the fibres n-GrdXE.