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), and semi-pseudometric spaces, in particular those constructed from a pseudometric spaces decorated with a privileged scale, which are those typically used for topological data analysis. Our construction proceeds by constructing a Grothendieck topology on the category $\mathcal{M}(c_X)$ of finite intersections of subsets of $(X,c_X)$ with non-empty interior, which is the natural generalization to closure spaces of the category $\mathcal{O}(X,τ)$ of open sets in a topological space. We continue by constructing the sheaf and Cech cohomologies on $\mathcal{M}(c_X)$, and standard constructions then give a spectral sequence between Cech cohomology and the sheaf cohomology on a site $(\mathcal{M}(c_X),J)$ which we use to identify several examples of non-topological closure spaces with non-trivial sheaf cohomology.