发表机构
Institute of Business Administration; University of Notre Dame（商业管理学院; 圣母大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出超图的三种同调理论（Γ、□、×），证明其互异且不同于嵌入同调，并引入扩展图离散同伦的超图同伦，发现仅□-同调具同伦不变性，且Γ-同调有独特结构行为。
AI 中文摘要
我们为超图引入了三种同调理论，即Γ-同调、□-同调和×-同调，并证明它们两两不同构，且与超图的嵌入同调不同。我们进一步引入了超图的同伦概念，该概念扩展了图的离散同伦理论。在所考虑的同调理论中，我们证明了□-同调在该同伦下保持不变，而Γ-同调和×-同调不满足同伦不变性。基于切除，我们还识别了Γ-同调所表现出的一种独特结构行为，这进一步将其与□-同调区分开来。
英文摘要
We introduce three homology theories for hypergraphs, namely $Γ$-homology, $\Box$-homology, and $\times$-homology, and show that they are pairwise non-isomorphic and distinct from the embedded homology of hypergraphs. We further introduce a notion of homotopy for hypergraphs that extends the discrete homotopy theory of graphs. Among the homology theories considered, we prove that $\Box$-homology is invariant under this homotopy, whereas $Γ$-homology and $\times$-homology fail to satisfy homotopy invariance. Based on excision, we also identify a distinctive structural behavior exhibited by $Γ$-homology that further differentiates it from $\Box$-homology.
