Cellular sheaf Laplacians on the set of simplices of symmetric simplicial set induced by hypergraph

TL;DR

This paper extends cellular sheaf Laplacians to symmetric simplicial sets derived from hypergraphs, providing a mathematical framework and formulas that unify these structures with existing simplicial complex Laplacians.

Contribution

It introduces a functor from hypergraphs to symmetric simplicial sets and defines cellular sheaf Laplacians on these sets, generalizing previous Laplacian constructions.

Findings

Formulas for cellular sheaf Laplacians on symmetric simplicial sets

Equivalence of Laplacians on ordered finite abstract simplicial complexes and induced sets

Unification of hypergraph structures with simplicial complex Laplacians

Abstract

We generalize cellular sheaf Laplacians on an ordered finite abstract simplicial complex to the set of simplices of a symmetric simplicial set. We construct a functor from the category of hypergraphs to the category of finite symmetric simplicial sets and define cellular sheaf Laplacians on the set of simplices of finite symmetric simplicial set induced by hypergraph. We provide formulas for cellular sheaf Laplacians and show that cellular sheaf Laplacian on an ordered finite abstract simplicial complex is exactly the ordered cellular sheaf Laplacian on the set of simplices induced by abstract simplicial complex.