A relative theory of universal central extensions

TL;DR

This paper develops a unified, relative framework for universal central extensions within semi-abelian categories, generalizing classical results and exploring conditions on the composition of such extensions.

Contribution

It introduces a relative theory of universal central extensions based on Birkhoff subcategories, unifying various classical and recent results in a single conceptual framework.

Findings

Unified classical and new results in a single framework

Identified conditions for composition of central extensions

Provided examples of categories satisfying or not satisfying these conditions

Abstract

Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. Thus we unify classical, recent and new results in one conceptual framework. The theory we develop is relative to a chosen Birkhoff subcategory of the category considered: for instance, we consider groups vs. abelian groups, Lie algebras vs. vector spaces, precrossed modules vs. crossed modules and Leibniz algebras vs. Lie algebras. We consider a fundamental condition on composition of central extensions and give examples of categories which do, or do not, satisfy this condition.