发表机构
Friedrich-Alexander Universität Erlangen-Nürnberg（弗里德里希-亚历山大大学埃尔兰根-纽伦堡分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明大小同调是长度滤过的相伴分次，将其应用于定量等式理论并给出条形稳定性估计与公理强度衡量方法，计算了四个不同度的示例。
AI 中文摘要
大小同调（Magnitude homology）按长度分级，不涉及持续性；其持续性细化版本则不知晓其条形（bar）的起止位置。我们证明二者是同一构造：对长度 nerve 按长度的亚水平集进行滤过，可得到持续性模，该滤过的相伴分次即为大小复形（magnitude complex）。长正合序列将二者关联，使每一方都获得了另一方所缺失的性质。大小同调可定位条形的临界值，因此分次计算能列出端点可能出现的长度，且在大小为δ的扰动下，条形在n次度下会获得(n+1)δ的稳定性估计；而经计算的扰动会使条形移动超过δ，故该因子无法省略。我们将此应用于定量等式理论，其自由代数是由句法构建的度量空间：理论的包含关系会诱导出表示单子（presenting monads）的态射，并给出带显式界的条形比较，因此该不变量可衡量公理强度。我们计算了四个示例，每个度各一个。
英文摘要
A quantitative equational theory $U$ reasons about terms that agree up to a numerical error. It presents a free algebra $T_UA$ over a metric space $A$ of generators, the terms of the syntax at the least distance the axioms derive, and that metric is its semantic content. We give a functorial invariant of it, the persistent magnitude homology of $T_UA$: a barcode where the module is tame, finite linear algebra where $T_UA$ is finite, Lipschitz in each degree. Magnitude homology is graded by length and knows nothing of persistence, its persistent refinement nothing of where its bars begin and end, yet 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 $δ$, and a computed perturbation shows that the factor cannot be dropped. An inclusion of theories induces a morphism of the presenting monads and, where the induced map is bijective and shortens no distance by more than $δ$, a comparison of barcodes under the same bound, so a barcode movement measures the metric-semantic strength of the added axioms. Four examples are computed, one in every degree.
CommentsCode available at https://codeberg.org/Jiren/PersHomAlg
