We study a combinatorial homotopy theory for small categories without a loop (loopfree categories) which is closely related to Whitehead's simple homotopy theory for regular CW-complexes with triangular cells. Quillen's theorem A and the nerve theorem for loopfree categories are considered from the viewpoint of the simple homotopy theory. Moreover, we extend the classical nerve theorem and discuss an application to the topological data analysis for data-sets lying in a non-convex field.

中文翻译：

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类别的简单同伦理论和神经定理

我们研究了无环的小类别（无环类别）的组合同伦理论，该理论与怀特海的规则CW复杂三角形细胞的简单同伦理论密切相关。从简单同伦理论的角度考虑了Quillen定理A和无环类别的神经定理。此外，我们扩展了经典神经定理，并讨论了在非凸域中对数据集进行拓扑数据分析的应用。