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

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.