Foundation of relative non-abelian homological algebra
Tamar Janelidze · Open University of Cape Town (University of Cape Town) · 2009
We consider pairs (C, E), where C is a pointed category and E a class of regular/normal epimorphisms in C, satisfying various exactness properties.The purpose of this thesis is:1. To introduce and study suitable notions of a relative homological and a relative semiabelian category.In the "absolute case", where E is the class of all regular epimorphisms in C, the pair (C, E) is relative homological/semi-abelian if and only if C is homological/semiabelian; that is, we obtain known concepts.Accordingly we extend known analysis of the axiom systems, and in particular show that suitable lists of "old style" and "new style" axioms are equivalent; this requires developing a relative version of what is usually called the calculus of relations.We then present various non-absolute examples, where these results can be applied.2. To formulate and prove relative versions of classical homological lemmas; this includes Five Lemma, Nine Lemma, and Snake Lemma.