Higher central extensions and cohomology

TL;DR

This paper develops a Galois-theoretic framework for understanding cohomology in semi-abelian categories, linking it to higher central extensions and dualities between homology and cohomology.

Contribution

It introduces a geometric and algebraic approach to higher central extensions, establishing a duality between internal homology and external cohomology in semi-abelian categories.

Findings

Cohomology classifies higher central extensions.

A duality between internal homology and external cohomology is established.

Results depend on geometric and algebraic perspectives of higher central extensions.

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.