By careful analysis of the embedding of a simplicial set into its image under Kan’s \(\mathop {\mathop {\mathsf {Ex}}^\infty }\) functor we obtain a new and combinatorial proof that it is a weak homotopy equivalence. Moreover, we obtain a presentation of it as a strong anodyne extension. From this description we can quickly deduce some basic facts about \(\mathop {\mathop {\mathsf {Ex}}^\infty }\) and hence provide a new construction of the Kan–Quillen model structure on simplicial sets, one which avoids the use of topological spaces or minimal fibrations.

中文翻译：

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Kan–Quillen模型结构的另一种方法

通过仔细分析在Kan的\（\ mathop {\ mathop {\ mathsf {\ mathsf {Ex}} ^ \ infty} \）函数下将单纯形集嵌入其图像中，我们获得了新的组合证明，它是弱同伦对等。此外，我们将其表示为强的止痛药扩展。从该描述中，我们可以快速得出有关\（\ mathop {\ mathop {\ mathsf {Ex}} ^ \ infty} \的一些基本事实，从而为简单集提供Kan-Quillen模型结构的新构造，其中一个避免使用拓扑空间或最小化纤维。