The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We address the question of universality, which asks for the largest possible stable distance on extended persistence diagrams, showing that a more discriminative variant of the bottleneck distance is universal. Our result applies more generally to settings where persistence diagrams are considered only up to a certain degree. We achieve our results by establishing a functorial construction and several characteristic properties of relative interlevel set homology, which mirror the classical Eilenberg--Steenrod axioms. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves on the real line by showing that the latter is not intrinsic, let alone universal. This particular result has the further implication that the interleaving distance of Reeb graphs is not intrinsic either.

翻译：扩展持久性图是分段线性函数的一个不变量，已知其在Cohen-Steiner、Edelsbrunner和Harer引入的瓶颈距离下对函数扰动具有稳定性。我们探讨通用性问题，即寻找扩展持久性图上最大的稳定距离，并证明瓶颈距离的一种更具区分性的变体是通用的。我们的结果更普遍地适用于仅考虑至特定维度的持久性图场景。通过建立相对层间同调的函数式构造及其若干特征性质（这些性质镜像了经典的Eilenberg-Steenrod公理体系），我们实现了上述结论。最后，我们通过证明实直线上的层交错距离非内蕴性（更遑论通用性），将其与瓶颈距离形成对比。该特定结果进一步暗示Reeb图的交错距离也非内蕴。