A Categorical Approach to Möbius Inversion via Derived Functors

Abstract

We develop a cohomological approach to Möbius inversion using derived functors in the enriched categorical setting. For a poset $P$ and a closed symmetric monoidal abelian category $\mathcal{C}$, we define Möbius 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 Möbius 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.

23 pages