Galois theory and the categorical Peiffer commutator

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