Baer sums in homological categories (Q868875)

The author proves that the description of the abstract Baer sum defined in any Barr exact context can be simplified when the ground category \(\mathcal C\) in question is moreover protomodular and can be reduced to the classical scheme concerning the exact sequences, then, this produces a unified treatment of the Baer theory in the context of any semi-abelian variety, in particular of any category of groups with multiple operators. The author also shows that the above result does hold in the category of Hausdorff groups and more generally any category of semi-abelian Hausdorff algebras. Finally, these processes lead to a generalized ``Euclide's Postulate'' and a five terms exact sequence.