# Galois theory and the categorical Peiffer commutator

## TL;DR

This paper explores the use of the Peiffer commutator to characterize central extensions and double central extensions of precrossed modules in semi-abelian categories, providing new homological formulas.

## Contribution

It introduces a novel application of the Peiffer commutator to characterize central extensions and derives Hopf formulas for homology in this context.

## Key findings

- Peiffer commutator characterizes central extensions in semi-abelian categories.
- Characterization of double central extensions using the same commutator.
- Derived Hopf formulas for second and third homology objects.

## Abstract

We show that the Peiffer commutator previously defined by Cigoli, Mantovani and Metere can be used to characterize central extensions of precrossed modules with respect to the subcategory of crossed modules in any semi-abelian category satisfying an additional property. We prove that this commutator also characterizes double central extensions, obtaining then some Hopf formulas for the second and third homology objects of internal precrossed modules.