We show that the category of internal groupoids in an exact Mal'tsev category is reflective, and in fact a Birkhoff subcategory of the category of simplicial objects. We then characterize the central extensions of the corresponding Galois structure, and show that regular epimorphisms admit a relative monotone-light factorization system in the sense of Chikhladze. We also draw some comparison with Kan complexes. By comparing the reflections of simplicial objects and reflexive graphs into groupoids, we exhibit a connection with weighted commutators (as defined by Gran, Janelidze and Ursini).

中文翻译：

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Mal'tsev 范畴中单纯对象的基本群状体

我们表明，精确 Mal'tsev 范畴中的内部 groupoids 范畴是反射性的，实际上是单纯对象范畴的 Birkhoff 子范畴。然后，我们描述了相应伽罗瓦结构的中心扩展，并表明正则同胚允许在 Chikhladze 意义上的相对单调光分解系统。我们还与 Kan 复合体进行了一些比较。通过将单纯对象和自反图的反射进行比较，我们展示了与加权交换子（如 Gran、Janelidze 和 Ursini 所定义的）的联系。