Higher Central Extensions via Commutators

We prove that all semi-abelian categories with the the Smith is Huq property satisfy the Commutator Condition (CC): higher central extensions may be characterised in terms of binary (Huq or Smith) commutators. In fact, even Higgins commutators suffice. As a consequence, in the presence of enough projectives we obtain explicit Hopf formulae for homology with coefficients in the abelianisation functor, and an interpretation of cohomology with coefficients in an abelian object in terms of equivalence classes of higher central extensions. We also give a counterexample against (CC) in the semi-abelian category of (commutative) loops. Introduction The concept of higher centrality is a cornerstone in the recent approach to homology and cohomology of non-abelian algebraic structures based on categorical Galois theory [5, 35]. Through higher central extensions, the Brown–Ellis–Hopf formulae [11, 14] which express homology objects as a quotient of commutators have been made categorical [17, 19, 20], which greatly extends their scope while simplifying the study of concrete cases (see, for instance, [13]). Higher central extensions are also essential in the study of relative commutators [22, 23] and are classified by cohomology groups [49]. To take full advantage of these results, sufficiently explicit characterisations of higher centrality are essential. On the one hand, the higher Hopf formulae are valid in any semiabelian category [39] with enough projectives, but these formulae only become concrete once the relevant concept of higher centrality is appropriately characterised, ideally in terms of classical binary commutators. Indeed, the main result of [20] says that in a semi-abelian monadic category A, for any n-presentation F of Z, Hn 1pZ, abq rFn, Fns ^ ™ iPn Kerpfiq LnrF s . (A) Coefficients are chosen in the abelianisation functor ab : AÑ AbpAq. Here Fn is the initial object of F and the fi are the initial arrows. The object rFn, Fns is the Huq commutator The first author was supported by CMUC/FCT (Portugal) and the FCT Grant PTDC/MAT/120222/2010 through the European program COMPETE/FEDER. The second author works as chargé de recherches for Fonds de la Recherche Scientifique–FNRS and would like to thank CMUC for its kind hospitality during his stays in Coimbra. Received by the editors 2012-03-26 and, in revised form, 2012-10-04. Published on 2012-10-08 in the volume of articles from CT2011. 2010 Mathematics Subject Classification: 18G50, 18G60, 18G15, 20J, 55N.