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Chapter 4 put full emphasize on the notion of homological category, in which all traditional homological lemmas hold. In a finitely cocomplete Mal’cev category, we obtained results concerning commutators. With always in mind the generic example of the category of groups, we present here the notion of semi-abelian category, introduced in : these categories constitute a context in which both aspects above can be developed. Unexpectedly, in this context, the notion of semi-direct product emerges naturally.
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