Commutators and central extensions in universal algebra

Abstract

We investigate internal groupoids and pseudogroupoids in varieties of universal algebras, and we give a new description of internal groupoids in congruence modular varieties. We then prove that in any congruence modular variety an algebraically central extension is categorically central. The converse implication being already known, it follows that there is a perfect agreement between these two notions in any congruence modular variety. This theorem extends various partial results in this direction proved, so far, for Ω-groups, for Maltsev varieties and for semi-abelian categories.