揭示了数量同调与持久同调的统一本质，打通了长度与稳定性分析的桥梁。
Magnitude Homology Is the Associated Graded of the Length Filtration
- 用长度滤波构造持久同调，其关联分次即为数量同调复形。
- 证明了在扰动δ下，第n度的条形码稳定界为(n+1)δ，且该因子不可改进。
- 适用于代数理论的度量自由代数，可量化公理体系的强弱程度。
数量同调按长度分次，但不涉及持久性；其持久性改进也不记录条形码的起止位置。我们证明二者实为同一构造：对长度神经复形按长度子水平集滤波，得到持久模，其关联分次即为数量复形。存在长正合列相互转换，双方均获对方所缺信息。数量同调定位条形码的关键值，使得分次计算能列出端点可能出现的长度；条形码在大小为δ的扰动下，第n度具有(n+1)δ的稳定性估计，且该因子不可省略。我们将其应用于定量等式理论，其自由代数是基于语法构建的度量空间：理论包含关系诱导呈现单子的态射，并给出条形码比较的显式界，使该不变量能度量公理强度。四个例子分别在每一度上完成计算。
原文摘要 · Abstract (English)
Magnitude homology is graded by length and knows nothing of persistence. Its persistent refinement knows nothing of where its bars begin and end. We show that the two are one construction: filtering the length nerve by sublevel sets of the length yields the persistence module, and the associated graded of that filtration is the magnitude complex. A long exact sequence exchanges them, and each side gains what it lacked. Magnitude homology locates the critical values of the barcode, so a graded computation lists the lengths at which an endpoint can occur, and the barcode acquires a stability estimate of $(n+1)δ$ in degree $n$ under a perturbation of size $δ$, while a computed perturbation moves a barcode by more than $δ$, so the factor cannot be dropped. We apply this to quantitative equational theories, whose free algebras are metric spaces built from syntax: an inclusion of theories induces a morphism of the presenting monads and a comparison of barcodes with an explicit bound, so the invariant measures axiomatic strength. Four examples are computed, one in every degree.
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