The denormalized 3×3 lemma

Abstract

The classical 3×3 lemma holds in any regular protomodular category C with a zero object. It is investigated here whether there is a “denormalized” version when the category no longer has a zero object, as, for instance, any slice category C/X of the category C , or any slice category of an abelian category. The answer is actually positive in the weaker context of regular Mal'cev categories.