Möbius Homology

2023-07-03

arXiv.org

AI Summary

Persistent diagrams in persistent homology lack structural interpretation. Method: We introduce Möbius homology—a novel homological theory for finite posets—by categorifying the Möbius inversion, thereby establishing systematic connections to Galois connections, representation theory, and algebraic topology; we prove a homological version of Rota’s Galois connection theorem and construct categorical invariants for persistent homology over arbitrary finite posets. Contributions/Results: (1) We show that the Euler characteristic of Möbius homology equals the Möbius inversion of the dimension function; (2) we uniformly interpret persistent diagrams as Euler characteristics over interval posets; (3) we derive several new structural results on persistent diagrams, substantially deepening the algebraic understanding of persistent homology.

Abstract

This paper introduces and develops M""obius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and M""obius functions is classical, we go further by establishing a direct connection between poset topology and M""obius inversions. In particular, we show that M""obius homology categorifies the M""obius inversion, as its Euler characteristic coincides with the M""obius inversion applied to the dimension function of the representation. We also present a homological version of Rota's Galois Connection Theorem, relating the M""obius homologies of two posets connected by a Galois connection. Our main application concerns persistent homology over general posets. We prove that, under a suitable definition, the persistence diagram arises as an Euler characteristic over a poset of intervals, and thus M""obius homology provides a categorification of the persistence diagram. This furnishes a new invariant for persistent homology over arbitrary finite posets. Finally, leveraging our homological variant of Rota's Galois Connection Theorem, we establish several results about the persistence diagram.

Problem

• Möbius Homology
• Persistent Homology
• Rota's Galois Connection

Innovation

• Möbius Homology
• Persistent Homology
• Rota's Galois Connection