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Categorical-algebraic methods in group cohomology

By Tim Van der Linden and Category Theory 2017


In the article [8], Janelidze introduced the concept of a double central exten- sion in order to analyse the Hopf formula for the third integral homology of a group [2]. Later it turned out that this “double extension” viewpoint on group homology may be extended to higher degrees, and at the same time generalised to the framework of semi-abelian categories [5]. Indeed, categorical Galois theory gives rise to the concept of an n-fold central extension (n ≥ 1), which is such that the higher Hopf formulae of [2, 3], suitably reinterpreted in terms of these higher central extensions, give an explicit description of the derived functors of any reflector from a semi-abelian variety to one of its subvarieties. In the particular case of the abelianisation reflection from the category of groups to the category of abelian groups, the Hopf formulae for integral group homology are thus regained. Central extensions do however also appear in group cohomology, in the interpretation of the second cohomology group with coefficients in a trivial Z-module A, which is one of the derived functors of the functor Hom(−, A). This result extends to semi-abelian categories [7] and to non-trivial coefficients (via the concept of a torsor [1]). On the other hand, in the abelian case there is Yoneda’s classical interpretation of these derived functors via classes of exact sequences of a certain fixed length [10]. In Barr-exact categories, the higher- dimensional torsors of [4] play essentially the same role. The aim of this talk is to explain how, in a semi-abelian context, these two developments are related. Through an equivalence between higher torsors (with trivial coefficients) and higher central extensions we obtain a duality, in a certain sense, between homology and cohomology [9, 6]. Even in the case of groups this viewpoint is new, but it is automatically valid as well for other non-abelian algebraic structures such as Lie algebras, crossed modules, associative algebras, and so on. In its most general version, the theory depends on some non-trivial re- cent developments in categorical algebra. Part of the talk focuses on these categorical-algebraic aspects: how questions in homological algebra naturally lead to categorical conditions and results. The need for further development of categorical algebra becomes particularly apparent in the case of cohomology with non-trivial coefficients. This case is much more complicated, because here the techniques of categorical Galois theory are no longer available. References [1] D. Bourn and G. Janelidze, Extensions with abelian kernels in protomodular categories, Georgian Math. J. 11 (2004), no. 4, 645–654. [2] R. Brown and G. J. Ellis, Hopf formulae for the higher homology of a group, Bull. Lond. Math. Soc. 20 (1988), no. 2, 124–128. [3] G. Donadze, N. Inassaridze, and T. Porter, n-Fold Cˇech derived functors and generalised Hopf type formulas, K-Theory 35 (2005), no. 3–4, 341–373. [4] J. Duskin, Simplicial methods and the interpretation of “triple” cohomology, Mem. Amer. Math. Soc., no. 163, Amer. Math. Soc., 1975. [5] T. Everaert, M. Gran, and T. Van der Linden, Higher Hopf formulae for homology via Galois Theory, Adv. Math. 217 (2008), no. 5, 2231–2267. [6] J. Goedecke and T. Van der Linden, On satellites in semi-abelian categories: Homology without projectives, Math. Proc. Cambridge Philos. Soc. 147 (2009), no. 3, 629–657. [7] M. Gran and T. Van der Linden, On the second cohomology group in semi-abelian categories, J. Pure Appl. Algebra 212 (2008), 636–651. [8] G. Janelidze, What is a double central extension? (The question was asked by Ronald Brown), Cah. Topol. G ́eom. Differ. Cat ́eg. XXXII (1991), no. 3, 191–201. [9] D. Rodelo and T. Van der Linden, Higher central extensions and cohomology, Adv. Math. 287 (2016), 31–108. [10] N. Yoneda, On Ext and exact sequences, J. Fac. Sci. Univ. Tokyo 1 (1960), no. 8, 507–576

Year: 2017
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