A Categorical Approach to M\"obius Inversion via Derived Functors

TL;DR

This paper introduces a cohomological framework for M"obius inversion using derived functors in enriched categories, providing a categorification that unifies combinatorics and homological algebra.

Contribution

It develops a novel cohomology theory for M"obius inversion in enriched categorical settings, extending classical combinatorial concepts through homological methods.

Findings

M"obius cohomology recovers classical inversion via Euler characteristic.

Categorical version of Rota's Galois Connection proved.

Unification of combinatorics and homological algebra achieved.

Abstract

We develop a cohomological approach to M\"obius inversion using derived functors in the enriched categorical setting. For a poset $P$ and a closed symmetric monoidal abelian category $\mathcal{C}$, we define M\"obius cohomology as the derived functors of an enriched hom functor on the category of $P$-modules. We prove that the Euler characteristic of our cohomology theory recovers the classical M\"obius inversion, providing a natural categorification. As a key application, we prove a categorical version of Rota's Galois Connection. Our approach unifies classical ideas from combinatorics with homological algebra.