The concept of a pair of compatible actions was introduced in the case of groups by Brown and Loday and in the case of Lie algebras by Ellis. In this article we extend it to the context of semi-abelian categories (that satisfy the Smith-is-Huq condition). We give a new construction of the Peiffer product, which specialises to the definitions known for groups and Lie algebras. We use it to prove our main result, on the connection between pairs of compatible actions and pairs of crossed modules over a common base object. We also study the Peiffer product in its own right, in terms of its universal properties, and prove its equivalence with existing definitions in specific cases.

中文翻译：

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半阿贝尔范畴中的兼容动作

Brown 和 Loday 在群的情况下和 Ellis 在李代数的情况下引入了一对相容动作的概念。在本文中，我们将其扩展到半阿贝尔范畴（满足 Smith-is-Huq 条件）的上下文。我们给出了 Peiffer 乘积的新构造，它专门研究群和李代数的定义。我们用它来证明我们的主要结果，即在一个公共基础对象上的兼容动作对和交叉模块对之间的连接。我们还研究了 Peiffer 乘积本身的通用属性，并证明了它在特定情况下与现有定义的等效性。