The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By contrast, the interleaving distance, introduced in the 2000s within the study of persistent homology, has become a foundational metric in topological data analysis. In this work, we show that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as a so-called Galois-edit distance. The key lies in clarifying a connection between the Galois connection and the interleaving distance, via the established relation between the interleaving distance and free presentations of persistence modules. In addition to offering new perspectives on the interleaving distance, we expect that our findings will facilitate the study of stability properties of invariants for multi-parameter persistence modules. As an application of the Galois-edit formulation of the interleaving distance, we present an alternative proof of the well-known bottleneck stability theorem.

翻译：编辑距离的概念可追溯至20世纪60年代，最初用于比较字符串，随后在计算机科学、计算生物学和应用拓扑学中通过多种变体得到了广泛应用。与之相对，21世纪初在持续同调研究中引入的交错距离已成为拓扑数据分析中的基础度量。本研究证明，有限表示的单参数和多参数持续模上的交错距离可表述为所谓的Galois-编辑距离。关键在于通过交错距离与持续模自由表示之间的已有关系，阐明Galois联系与交错距离之间的关联。除为交错距离提供新视角外，我们预期该发现将有助于研究多参数持续模不变量的稳定性性质。作为交错距离的Galois-编辑表述的应用，我们给出了著名的瓶颈稳定性定理的另一种证明。