In the context of internal crossed modules over a fixed base object in a given semi-abelian category, we use the non-abelian tensor product in order to prove that an object is perfect (in an appropriate sense) if and only if it admits a universal central extension. This extends results of Brown and Loday (Topology 26(3):311–335, 1987, in the case of groups) and Edalatzadeh (Appl Categ Struct 27(2):111–123, 2019, in the case of Lie algebras). Our aim is to explain how those results can be understood in terms of categorical Galois theory: Edalatzadeh’s interpretation in terms of quasi-pointed categories applies, but a more straightforward approach based on the theory developed in a pointed setting by Casas and Van der Linden (Appl Categ Struct 22(1):253–268, 2014) works as well.

中文翻译：

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通过非阿贝尔张量积的内部交叉模块的通用中心扩展

在给定半阿贝尔范畴中固定基础对象上的内部交叉模块的上下文中，我们使用非阿贝尔张量积来证明一个对象是完美的（在适当的意义上）当且仅当它承认一个通用中央扩展。这扩展了 Brown 和 Loday（Topology 26(3):311–335, 1987, 对于群）和 Edalatzadeh（Appl Categ Struct 27(2):111–123, 2019，对于李代数）的结果. 我们的目的是解释如何根据分类伽罗瓦理论来理解这些结果：Edalatzadeh 对准点类别的解释适用，但基于 Casas 和 Van der Linden 在有针对性的环境中开发的理论的更直接的方法（ Appl Categ Struct 22(1):253–268, 2014) 也适用。