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

Alex Elchesen; Amit Patel

Home/Archive/arXiv

arXiv · 2411.04362

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

Alex Elchesen·Amit Patel

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.AT math.CO math.CT math.KT

Keep this discovery

BibTeX RIS

Alex Elchesen, Amit Patel. 2024-11-07. A Categorical Approach to M\"obius Inversion via Derived Functors. https://arxiv.org/abs/2411.04362

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Homogeneous Milnor fibers and Kato--Matsumoto bounds via simplicial multiwedges

For every $n\geq 3$ and $s\geq 2$, we construct a homogeneous polynomial of degree $n(n+1)/2$ whose Milnor fiber is exactly $2s$-connected and whose rational cohomology contains a strictly defined nontrivial $n$-fold Massey product on classes of degree $2s+1$, implying that the Milnor fiber is non-formal, while attaining the Kato--Matsumoto connectivity bound. Our construction is based on the simplicial multiwedges of the nerve complexes of simple polytopes introduced by Limonchenko, combined with Suciu's realization of weighted homogeneous Milnor fibers. We thereby answer two problems posed by Suciu.

math.AT↗

arXiv2026

The homotopy types of directed path and trace spaces

We construct a saturated directed space with a Hausdorff $\Delta$-generated underlying space and two distinct points such that the trace space between them is homeomorphic to a square, whereas the directed path space has a nontrivial fundamental group. In particular, the canonical quotient map is not a weak homotopy equivalence. The same conclusion holds for regular directed paths modulo increasing homeomorphisms.

math.AT↗

arXiv2026

Moduli spaces of geometric functorial field theories

We develop tools to compute moduli spaces of geometric functorial field theories as mapping spaces of equivariant simplicial presheaves. Given a d-dimensional geometric structure F, presented as a presheaf on the site of smooth families of d-manifolds, we define its Cartesian realization, which is an O(d)-equivariant simplicial presheaf on the site of Cartesian spaces. We use Cartesian realizations to present the moduli space of functorial field theories with geometric structure F as a mapping space between O(d)-equivariant simplicial presheaves. In a companion paper, we use this result to compute the moduli space of smooth one-dimensional oriented Riemannian functorial field theories valued in an arbitrary smooth symmetric monoidal infinity-category.

math.AT↗