Geodesic
Peiffer product and Peiffer commutator for internal pre-crossed modules
Homology, homotopy, and applications, Tome 19 (2017) no. 1, pp. 181-207.

Voir la notice de l'article provenant de la source International Press of Boston

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 $B$.
DOI : 10.4310/HHA.2017.v19.n1.a10
Classification : 08C05, 18A30, 18C05, 18D35, 18G50, 18G55
Keywords: crossed module, Peiffer commutator, semi-abelian category 
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     title = {Peiffer product and {Peiffer} commutator for internal pre-crossed modules},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2017.v19.n1.a10/}
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Alan S. Cigoli; Sandra Mantovani; Giuseppe Metere. Peiffer product and Peiffer commutator for internal pre-crossed modules. Homology, homotopy, and applications, Tome 19 (2017) no. 1, pp. 181-207. doi : 10.4310/HHA.2017.v19.n1.a10. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2017.v19.n1.a10/

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