M\"obius Homology

TL;DR

This paper develops M"obius homology, a new homology theory for poset representations, which categorifies M"obius inversion and offers novel insights into persistent homology over general posets.

Contribution

It introduces M"obius homology, linking poset topology with homological algebra, and applies it to categorify persistence diagrams in persistent homology.

Findings

M"obius homology categorifies M"obius inversion.

Persistence diagrams can be viewed as Euler characteristics in a poset of intervals.

New invariants for persistent homology over arbitrary finite posets.

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.