The cohomological comparison arising from the associated abelian object

We make explicit some conditions on a semi-abelian category D such that, for any abelian group A in D and any object Y in D, the cohomology group homomorphisms with coefficients in A, induced by the inclusion of the abelian objects of D at the level of the slice category D/Y, are actually isomorphisms. These conditions hold in particular when D is the category Gp of groups, and this allows us to give a new insight on the Eilenberg-Mac Lane cohomology of groups. They hold also when D is the category K-Lie of Lie-algebras.

💡 Research Summary

The paper investigates a precise categorical setting in which cohomology groups computed in a semi‑abelian category D agree with those computed after restricting to the abelian objects of a slice D/Y. The author introduces two structural conditions on D: (i) D must be “peri‑abelian”, meaning that for every regular epimorphism f : X → Y the abelianisation functor Ab commutes with f; equivalently, the kernel of f is an abelian object and the process of taking kernels and abelianisation are compatible. (ii) Every object Y of D must have projective dimension 1, i.e. there exists a short exact sequence 0 ← Y ← P₀ ← P₁ ← 0 with P₀ and P₁ projective. Under these hypotheses the natural map induced by the inclusion of the full subcategory of abelian objects \