Geodesic
Higher central extensions via commutators
Theory and applications of categories, CT2011, Tome 27 (2012), pp. 189-209

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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.

Publié le :
Classification : 18G50, 18G60, 18G15, 20J, 55N
Keywords: Higgins, Huq, Smith commutator, higher central extension, semi-abelian, exact Mal'tsev category, Hopf formula, (co)homology 
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     author = {Diana Rodelo and Tim Van der Linden},
     title = {Higher central extensions via commutators},
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     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2012_27_a8/}
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Diana Rodelo; Tim Van der Linden. Higher central extensions via commutators. Theory and applications of categories, CT2011, Tome 27 (2012), pp. 189-209. http://geodesic.mathdoc.fr/item/TAC_2012_27_a8/