Higher central extensions and Hopf formulae

Higher extensions and higher central extensions, which are of importance to non-abelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained.

💡 Research Summary

The paper develops a systematic theory of higher extensions and higher central extensions within the framework of non‑abelian homological algebra, and uses this theory to give a direct proof of the invariance of the higher Hopf formulae. After recalling the basic setting of semi‑abelian categories and a chosen Birkhoff subcategory 𝔅 together with its reflector I:𝔄→𝔅, the authors introduce the notion of an n‑fold extension. An n‑fold extension is a ladder of regular epimorphisms
Xₙ → Xₙ₋₁ → … → X₁ → X₀,
each possessing a kernel, and the whole ladder is required to be a regular diagram. This generalises the classical one‑dimensional extension and provides the basic object for higher homological constructions.

The central contribution is the definition of a higher central extension: an n‑fold extension whose each kernel commutes with the 𝔅‑normal subobjects, i.e. each stage is 𝔅‑central. Using the commutator calculus associated with the reflector I, the authors construct a canonical “centralisation” functor C that assigns to any n‑fold extension its maximal 𝔅‑central part. They prove three fundamental properties: (1) existence and uniqueness of C(e) for any n‑fold extension e, (2) that the collection of higher central extensions forms a full reflective subcategory of the category of regular n‑fold extensions, and (3) closure under pullbacks, composition and finite coproducts, showing that the subcategory is itself semi‑abelian.

With this machinery in place, the paper turns to the higher Hopf formulae, which express the n‑th homology Hₙ(A) of an object A in terms of a projective presentation P and the kernels appearing in a higher central extension C. The classical Hopf formula for H₁(A) is recovered when n = 1. For n ≥ 2 the authors obtain
Hₙ(A) ≅ Ker(Cₙ) /