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Normal monomorphisms in the sense of Bourn describe the equivalence classesof an internal equivalence relation. Although the definition is given in thefairly general setting of a category with finite limits, later investigationson this subject often focus on protomodular settings, where normality becomes aproperty. This paper clarifies the connections between internal equivalencerelations and Bourn-normal monomorphisms in regular Mal'tesv categories withpushouts of split monomorphisms along arbitrary morphisms, whereas a fulldescription is achieved for quasi-pointed regular Mal'tsev categories withpushouts of split monomorphisms along arbitrary morphisms.

Bourn-normal monomorphisms in regular Mal'tsev categories