Homology, Homotopy and Applications

Volume 19 (2017)

Number 1

Peiffer product and Peiffer commutator for internal pre-crossed modules

Pages: 181 – 207

DOI: https://dx.doi.org/10.4310/HHA.2017.v19.n1.a10


Alan S. Cigoli (Institut de recherche en mathématique et physique, Université catholique de Louvain, Louvain-la-neuve, Belgium; and Dipartimento di Matematica, Università degli Studi di Milano, Italy)

Sandra Mantovani (Dipartimento di Matematica, Università degli Studi di Milano, Italy)

Giuseppe Metere (Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Italy)


In this work we introduce the notions of Peiffer product and Peiffer commutator of internal pre-crossed modules over a fixed object $B$, extending the corresponding classical notions to any semi-abelian category $\mathcal{C}$. We prove that, under mild additional assumptions on $\mathcal{C}$, crossed modules are characterized as those pre-crossed modules $X$ whose Peiffer commutator $\langle X, X \rangle$ is trivial. Furthermore we provide suitable conditions on $\mathcal{C}$ (fulfilled by a large class of algebraic varieties, including among others groups, associative algebras, Lie and Leibniz algebras) under which the Peiffer product realizes the coproduct in the category of crossed modules over $B$.


crossed module, Peiffer commutator, semi-abelian category

2010 Mathematics Subject Classification

08C05, 18A30, 18C05, 18D35, 18G50, 18G55

Published 6 June 2017