Galois theory and categorical Peiffer commmutator

Cigoli, Alan Stefano;Duvieusart, Arnaud;Gran, Marino;Mantovani, Sandra
(2020) Homology, Homotopy and Applications — Vol. 22, n° 2, p. 323-346 (2020)

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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.
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Cigoli, A. S., Duvieusart, A., Gran, M., & Mantovani, S. (2020). Galois theory and categorical Peiffer commmutator. Homology, Homotopy and Applications, 22(2), 323-346. https://doi.org/10.4310/HHA.2020.v22.n2.a20 (Original work published 2020)