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A Categorical Approach to Möbius Inversion via Derived Functors

Alex Elchesen, Amit Patel

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.

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 and a closed symmetric monoidal abelian category , we define Möbius cohomology as the derived functors of an enriched hom functor on the category of -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.

Paper Structure

This paper contains 31 sections, 21 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Previous Work
  3. Our contributions
  4. Outline
  5. Categorical Preliminaries
  6. Adjunctions
  7. Abelian Categories
  8. Closed Monoidal Categories
  9. Abelian Symmetric Monoidal Categories
  10. Ends
  11. Injective Resolutions
  12. Grothendieck Ring
  13. Poset Modules
  14. First Constructions
  15. Enriched Hom
  16. ...and 16 more sections

Key Result

Proposition 2.1

For any closed symmetric monoidal category $(\mathcal{C}, \otimes, {\mathbf{1}})$ and for all objects $B$ in $\mathcal{C}$, we have ${\mathcal{H} \mathrm{om}}_\mathcal{C}({\mathbf{1}}, B) \cong B.$

Theorems & Definitions (53)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • Proposition 3.7
  • proof
  • ...and 43 more