对应模与持久层：单参数持久同调的统一观点
Correspondence Modules and Persistence Sheaves: A Unifying Perspective on One-Parameter Persistent Homology
摘要：
我们开发了一个统一的框架，用于治疗各种持久的同源架构使用的概念对应模块。在这个公式中，向量空间之间的态射由部分线性关系给出，而不是线性映射。在一维的情况下，除其他外，这允许我们：（i）将持久化模块和zigzag模块视为相同类型的代数对象;（ii）给出连续参数上zigzag结构的分类公式;以及（iii）构造与几何信息更丰富的空间和映射相关联的条形码。单参数持久性的结构分析进行了对应模块，产生层状结构，称为持久性层的部分的水平。在一定的驯服性假设下，我们证明了持久层和对应模的区间分解定理，以及从区间分解得到的持久图的等距定理。应用包括：（a）一个Mayer-Vietoris序列，它将子水平集滤子和超水平集滤子的持续同调与实值函数的水平集同调模联系起来;（b）沿负斜线构造2-参数持续模的切片。
英文摘要：
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this formulation, morphisms between vector spaces are given by partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures over a continuous parameter; and (iii) construct barcodes associated with spaces and mappings that are richer in geometric information. A structural analysis of one-parameter persistence is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions. Applications include: (a) a Mayer-Vietoris sequence that relates the persistent homology of sublevelset filtrations and superlevelset filtrations to the levelset homology module of a real-valued function and (b) the construction of slices of 2-parameter persistence modules along negatively sloped lines.
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