A combinatorial approach to categorical M\"obius inversion and pseudoinversion

TL;DR

This paper presents a combinatorial interpretation of M"obius inversion and pseudoinversion using path sums in digraphs, extending previous theorems and applying to metric space magnitude and finite categories.

Contribution

It introduces a novel combinatorial framework for M"obius inversion and pseudoinversion, generalizing prior results and linking to metric space and category theory.

Findings

Provides a path-based formula for M"obius coefficients

Extends the framework to Moore-Penrose pseudoinverses

Offers a new expression for metric space magnitude

Abstract

We use Cramer's formula for the inverse of a matrix and a combinatorial expression for the determinant in terms of paths of an associated digraph (which can be traced back to Coates) to give a combinatorial interpretation of M\"obius inversion whenever it exists. Every M\"obius coefficient is a quotient of two sums, each indexed by certain collections of paths in the digraph. Our result contains, as particular cases, previous theorems by Hall (for posets) and Leinster (for skeletal categories whose idempotents are identities). A byproduct is a novel expression for the magnitude of a metric space as sum over self-avoiding paths with finitely many terms. By means of Berg's formula, our main constructions can be extended to Moore-Penrose pseudoinverses, yielding an analogous combinatorial interpretation of M\"obius pseudoinversion and, consequently, of the magnitude of an arbitrary finite category.