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We introduce relative homological and weakly homological categories (C, E), where " relative " refers to a distinguished class E of normal epimorphisms in C. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative versions of various homological lemmas can be proved in a… (More)
The direction of abelian relative homological algebra based on relative exactness properties was initiated by N. Yoneda, whose quasi-abelian categories  can be seen as additive categories equipped with a distinguished class E of epimorphisms, so that the short exact sequences K → A → B with A → B in E have the same properties as all short exact sequences… (More)
The purpose of this paper is to prove a new, incomplete-relative, version of Non-abelian Snake Lemma, where " relative " refers to a distinguished class of normal epimorphisms in the ground category, and " incomplete " refers to omitting all complete-ness/cocompleteness assumptions not involving that class.