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

Antonio Rieser

TL;DR

The paper develops a unified sheaf-theoretic framework for Čech closure spaces, unifying topological spaces, graphs, digraphs, quivers, and mesoscopic metric spaces under a single formalism. It constructs a site on the category $\mathcal{M}_{c_X}$ using interior covers, induces a Grothendieck topology $J_X$, and defines both sheaf and Čech cohomology in this closure-space setting. Key contributions include the first comprehensive treatment of sheaves on directed graphs via closure spaces, the establishment of adjunctions for sheaf operations in this context, and the demonstration that non-topological closure spaces can have nontrivial cohomology, notably in dimension two. This framework offers a flexible tool for applying algebraic topology to discrete and data-driven structures, with potential impact on topological data analysis and related areas by enabling cohomological methods directly on graphs, quivers, and mesoscopic spaces.

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 $\mathcal{O}(X,τ)$ 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.

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

TL;DR

The paper develops a unified sheaf-theoretic framework for Čech closure spaces, unifying topological spaces, graphs, digraphs, quivers, and mesoscopic metric spaces under a single formalism. It constructs a site on the category using interior covers, induces a Grothendieck topology , and defines both sheaf and Čech cohomology in this closure-space setting. Key contributions include the first comprehensive treatment of sheaves on directed graphs via closure spaces, the establishment of adjunctions for sheaf operations in this context, and the demonstration that non-topological closure spaces can have nontrivial cohomology, notably in dimension two. This framework offers a flexible tool for applying algebraic topology to discrete and data-driven structures, with potential impact on topological data analysis and related areas by enabling cohomological methods directly on graphs, quivers, and mesoscopic spaces.

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 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 of finite intersections of subspaces of with non-empty -interior, which is the natural generalization to closure spaces of the category of open sets in a topological space. We continue by constructing the sheaf and Cech cohomologies on , and we then identify examples of non-topological closure spaces induced by graphs with non-trivial sheaf cohomology, in particular in dimension two.

Paper Structure

This paper contains 10 sections, 18 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Čech Closure Spaces
  3. Sheaves on Čech Closure Spaces
  4. Sites on the category $\mathcal{M}_{c_X}$
  5. The Grothendieck Topology on $\mathcal{M}_{c_X}$
  6. Sheaves on the Site $\tilde{\mathcal{\mathcal{M}}}_{c_X}$
  7. Operations on sheaves and presheaves
  8. Sheaf Cohomology on $\tilde{\mathcal{\mathcal{M}}}_{c_X}$
  9. Sheaf Cohomology
  10. Čech Cohomology

Key Result

Proposition 2.3

Let $(X, c_X)$ be a closure space. The interior operator $i_X : \mathcal{P} (X) \to \mathcal{P} (X)$ satisfies the following

Theorems & Definitions (53)

  • Definition 2.1
  • Definition 2.3
  • Proposition 2.3: Cech_1966, 14 A.11
  • Corollary 2.4
  • Corollary 2.5
  • proof
  • Definition 2.4
  • Proposition 2.6: Cech_1966, Theorem 17.A.9(a,b)
  • Definition 2.5
  • Example 2.7
  • ...and 43 more