Möbius Homology

TL;DR

The paper defines Möbius homology for representations of finite posets in abelian categories and proves its Euler characteristic recovers Möbius inversion of the dimension function, providing a homological categorification of Möbius inversion. It establishes a homological version of Rota’s Galois Connection, enabling comparisons of Möbius data across posets linked by Galois connections and yielding computational tools. A central application is persistent homology over general posets: the persistence diagram arises as an Euler characteristic over the interval poset, and persistent Möbius homology offers a finer invariant than the standard diagram, with dimension bounds and stability-informed structure derived from the Galois framework. The work thereby generalizes persistence to arbitrary finite posets, connects incidence-algebra Möbius inversions to cosheaf homology, and provides practical theorems for bounding and computing persistent invariants.

Abstract

This paper introduces and develops Möbius homology, a homology theory for representations of finite posets into abelian categories. Although the connection between poset topology and Möbius functions is classical, we go further by establishing a direct connection between poset topology and Möbius inversions. In particular, we show that Möbius homology categorifies the Möbius inversion, as its Euler characteristic coincides with the Möbius inversion applied to the dimension function of the representation. We also present a homological version of Rota's Galois Connection Theorem, relating the Möbius 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 Möbius 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.

Figures (10)

• Figure 1: Poset $P$ along with two $P$-modules $M$ and $N$ valued in ${\mathrm{Vec}}$.
• Figure 2: Poset $P$ along with two $P$-modules $M$ and $N$ valued in $\mathrm{End}(\mathbb{C})$.
• Figure 3: Module over face poset for the cusp map.
• Figure 4: Modules of subgroups.
• Figure 5: $M_\mu$ is a $P$-module in ${\mathrm{Vec}}$ for every $\mu \in {\mathsf{k}}$.

Theorems & Definitions (76)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Example 2.6
• Example 2.7
• Example 2.8
• Definition 2.9
• Proposition 2.10