On low-dimensional homology in categories
In the mid 1960s, two very useful results were proved linking homology with variety theory. In papers by \textit{J. Stallings} [J. Algebra 2, 170--181 (1965; Zbl 0135.05201)] and \textit{U. Stammbach} [Math. Z. 94, 157--177 (1966; Zbl 0144.26001)], it was shown that a short exact sequence of groups yielded a 5-term exact sequence involving the homology and a relative abelianisation term, and in Stalling's paper, it was shown that a homomorphism \(f : G\to H\) of groups satisfying a low-dimensional homological condition gave an isomorphism on the quotients \(G/G^n \to H/H^n\), where \(G^n\), \(H^n\) are the \(n\)th terms of the lower central series of \(G\) and \(H\), respectively. This was thus a result relative to the variety of abelian groups and was also extended to nilpotent groups of given class. Later work proved analogous results relative to general varieties, then to varieties of \(\Omega\)-groups (i.e., groups with additional operations) and in [Theory Appl. Categ. 12, 1--33 (2004; Zbl 1065.18011)], \textit{T. Everaert} and \textit{T. van der Linden} extended it to any semi-abelian category with enough projectives. Other useful generalisations were known, however, which were not covered by this general result. \textit{D. Conduché} and \textit{G. J. Ellis} [J. Algebra 123, No.~2, 327--335 (1989; Zbl 0698.18008)], for instance, had proved the corresponding result for pre-crossed modules over a fixed group \(B\) relative to the corresponding subcategory of crossed modules. This paper adopts an axiomatic approach unifying the results in the context of a quasi-pointed exact protomodular category with enough projectives relative to a fixed Birkhoff subcategory. This generality allows proofs of interesting new instances of homology relative to a variety to be derived.
scientific article; zbMATH DE number 5125924