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 $langle X, X angle$ 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 $B$.
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机译：在本文中，我们介绍了固定对象$ B $上内部预交叉模块的Peiffer乘积和Peiffer换向器的概念，并将相应的经典概念扩展到了任何半阿贝尔类$ mathcal {C} $。我们证明，在对$ mathcal {C} $的轻微附加假设下，交叉模块的特征是那些预先交叉的模块$ X $，其Peiffer换向器$ langle X，X rangle $是平凡的。此外，我们在$ mathcal {C} $（由一大类代数变体完成，包括关联代数，Lie和Leibniz代数等组）提供合适的条件，在这些条件下，Peiffer产品实现交叉模块类别中的副产品超过$ B $。
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