Higher commutator conditions for extensions in Mal'tsev categories

TL;DR

This paper introduces higher commutator conditions to characterize central extensions in Mal'tsev categories, generalizing classical results and providing new insights into the structure of various algebraic categories.

Contribution

It defines a Galois structure on pairs of equivalence relations and characterizes central extensions using higher commutator conditions, extending previous theories.

Findings

Characterization of central extensions via higher commutator conditions

Generalization of abelianization and internal reflexive graph results

Applications to groups, precrossed modules, and compact groups

Abstract

We define a Galois structure on the category of pairs of equivalence relations in an exact Mal'tsev category, and characterize central and double central extensions in terms of higher commutator conditions. These results generalize both the ones related to the abelianization functor in exact Mal'tsev categories, and the ones corresponding to the reflection from the category of internal reflexive graphs to the subcategory of internal groupoids. Some examples and applications are given in the categories of groups, precrossed modules, precrossed modules of Lie algebras, and compact groups.