Grothendieck Topologies and Sheaf Theory for Data and Graphs: An Approach Through Cech Closure Spaces

TL;DR

This paper develops a unified sheaf theory framework for various spaces including graphs, simplicial complexes, and metric spaces using Cech closure spaces, extending classical topological concepts to broader contexts.

Contribution

It introduces a Grothendieck topology on closure spaces and constructs sheaf and Cech cohomologies applicable to non-topological spaces like graphs and quivers.

Findings

Non-trivial sheaf cohomology in graph-induced closure spaces

Extension of sheaf theory to finite simplicial complexes and graphs

Identification of cohomological properties in dimension two

Abstract

We initiate the study of sheaves on Cech closure spaces, providing a new, unified approach to sheaf theory on many of the major classes of spaces of interest to applications: topological spaces, finite simplicial complexes (seen as $T_0$ topological spaces), graphs and digraphs (both seen as closure spaces), quivers (seen as a pair of closure spaces), and metric spaces decorated with a privileged scale, the latter of which are widely used in topological data analysis. Our construction proceeds by constructing a Grothendieck topology on the category $\mathcal{M}_{c_X}$ of finite intersections of subspaces of $(X,c_X)$ with non-empty $c_X$-interior, which is the natural generalization to closure spaces of the category $\mathcal{O}(X,\tau)$ of open sets in a topological space. We continue by constructing the sheaf and Cech cohomologies on $\mathcal{M}_{c_X}$, and we then identify examples of non-topological closure spaces induced by graphs with non-trivial sheaf cohomology, in particular in dimension two.