Baer sums in homological categories

We give a unified treatment of the Baer sums in the context of efficiently homological categories which, on the one hand, contains any category of groups with multiple operators and more generally any semi-abelian variety and, on the other hand, the category of Hausdorff groups and more generally any category of semi-abelian Hausdorff algebras. This gives rise to a generalized “Euclideʹs Postulate” and a five terms exact sequence.