A central extension is a regular epimorphism in a Barr exact category C satisfying suitable conditions involving a given Birkhoff subcategory of C (joint work with G. M. Kelly, 1994). In this paper we take C to be the category of (not-necessarily-unital) algebras over a (unital) commutative ring and consider central extensions with respect to the category of commutative algebras. We propose a new approach that avoids the intermediate notion of central extension due to A. Fröhlich in showing that $\alpha:A\to B$ is a central extension if and only if $aa'=a'a$ for all $a,a'\in A$ with $\alpha(a')=0$. This approach motivates introducing what we call weakly action representable categories, and we show that such categories are always action accessible. We also make remarks on what we call initial weak representations of actions and formulate several open questions.
Keywords: central extension, split extension, semi-abelian category, action accessible category, action representable category, weakly action representable category
2020 MSC: 18E13, 18E50, 16S70
Theory and Applications of Categories, Vol. 38, 2022, No. 36, pp 1395-1408.