Cellular Sheaves on Higher-Dimensional Structures

TL;DR

This paper extends cellular sheaf theory to higher-dimensional simplicial complexes, developing geometric and algebraic frameworks for modeling complex multi-way interactions in topological data analysis.

Contribution

It introduces methods for constructing non-trivial cellular sheaves on higher-dimensional structures, unifying geometric and algebraic approaches for advanced topological modeling.

Findings

Sheaf Laplacians recover classical ANM Hessian matrices

Higher-dimensional sheaf Laplacians encode multi-way interactions

Unified geometric and algebraic frameworks for complex systems

Abstract

Defining cellular sheaves beyond graph structures, such as on simplicial complexes containing higher-dimensional simplices, is an essential and intriguing topic in topological data analysis (TDA) and the development of sheaf neural networks. In this paper, we explore methods for constructing non-trivial cellular sheaves on spaces that include structures of dimension greater than one. This extends the focus from 0- or 1-dimensional components, such as vertices and edges, to elements like triangles, tetrahedra, and other higher-dimensional simplices within a simplicial complex. We develop a unified framework that incorporates both geometric and algebraic approaches to modeling such complex systems using cellular sheaf theory. Motivated by the geometric and physical insights from anisotropic network models (ANM), we first introduce constructions that define sheaf structures whose 0-th sheaf Laplacians recover classical ANM Hessian matrices. The higher-dimensional sheaf Laplacians in this setting encode additional patterns of multi-way interactions. In parallel, we propose an algebraic framework based on commutative algebra and ringed spaces, where sheaves of ideals and modules are used to define sheaf structures in a combinatorial and algebraically grounded manner. These two perspectives -- the geometric-physical and the algebraic -- offer complementary strengths and together provide a versatile framework for encoding structural relationships and analyzing multi-scale data over simplicial complexes.