从旧范畴模型到新范畴模型：通过开覆盖
From old categorical models to new ones through open covers
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- Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学）
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中文总结 AI 辅助
本文通过用相容范畴模型替代开覆盖的交集，并利用Grothendieck构造组装局部模型，推广了神经定理，得到有限无圈范畴，其分类空间保持原空间的弱同伦型，且构造是递归的。
中文摘要 AI 辅助
神经定理为拓扑空间的同伦型提供了基于合适开覆盖的组合模型。我们将这种局部到整体的方法进行推广，用相容的范畴模型替代覆盖的交集。这些局部模型通过Grothendieck构造进行组装，产生一个小范畴，其分类空间的弱同伦型与原空间相同。在适当的假设下，所得范畴是有限且无圈的。我们发展了该构造的两个版本。第一个版本以覆盖的非空有限交集的通常偏序集为指标。第二个版本使用成员偏序集，受Sancho de Salas的有限空间构造启发，该构造仅保留由空间中的点实现的成员模式，并针对点有限覆盖定义。经典神经定理及其分量变体作为特例被恢复。该构造是递归的：局部片段的范畴模型，连同表示其包含的函子，可以组合起来产生更复杂空间的范畴模型。
英文摘要
The nerve theorem provides a combinatorial model for the homotopy type of a topological space from a suitable open cover. We extend this local-to-global approach by replacing the intersections of the cover with compatible categorical models. These local models are assembled through the Grothendieck construction, yielding a small category whose classifying space has the weak homotopy type of the original space. Under suitable hypotheses, the resulting category is finite and acyclic. We develop two versions of this construction. The first is indexed by the usual poset of nonempty finite intersections of the cover. The second uses the membership poset, inspired by the finite-space construction of Sancho de Salas, which retains only the membership patterns realized by points of the space and is defined for point-finite covers. The classical nerve theorem and its componentwise variant are recovered as particular cases. The construction is recursive: categorical models of local pieces, together with functors representing their inclusions, can be combined to produce categorical models of more complex spaces.
