Higher central extensions and cohomology

Abstract

Abstract. We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily high degrees. This allows us to obtain a duality, in a certain sense, between "internal" homology and "external" cohomology in semi-abelian categories. These results depend on a geometric viewpoint of the concept of a higher central extension, as well as the algebraic one in terms of commutators.

The Yoneda isomorphism commutes with homology

“…The characterisation of central extensions on which Definition 5.3 is based works if binary Higgins commutators suffice to express higher centrality. This happens [46] whenever the Smith commutator of equivalence relations [48] agrees with the Huq commutator of normal subobjects [30]. Semiabelian categories satisfying this Smith is Huq condition (SH) include pointed strongly protomodular varieties [4] and categories of interest [43].…”

“…Here we explain a semiabelian analogue of Yoneda's interpretation of Ext; see [47] for details. It is based on categorical Galois theory and the concept of higher central extension.…”

“…'Congruence' generates an equivalence relation on central n-extensions under A and over X; let Centr n pX, Aq denote the resulting set of equivalence classes. The main results of [47,46], based on torsor theory, combined provide a natural isomorphism φ : Ext n pX, Aq -ÝÑ Centr n pX, Aq whenever X is a semiabelian category with enough projectives satisfying the Smith is Huq condition (see 2.5).…”

“…Subsequently, work toward defining congruence classes of higher central extensions which correspond to cohomology required a refinement of the concept. Such a refinement is proposed in [47]. Accordingly, we now use the term 'n-fold central extension' exclusively for a diagram as defined in 5.3.…”

“…When R is an abelian subvariety of a semi-abelian variety X satisfying the socalled Smith is Huq condition 2.5, recent work in [47] tells us that for M in R, the group Ext n pX, M q is (naturally) isomorphic to the group of equivalence classes of central n-extensions under M and over X. Hence Yoneda's lemma yields the existence of a universal central n-extension under L n T pXq and over X; see 6.7.…”