Grothendieck Topologies and Sheaf Theory for Data and Graphs: An Approach Through Cech Closure Spaces
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866912657701863424 |
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| author | Rieser, Antonio |
| author_facet | Rieser, Antonio |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_13867 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Grothendieck Topologies and Sheaf Theory for Data and Graphs: An Approach Through Cech Closure Spaces Rieser, Antonio Algebraic Topology 55N30, 55N31 (Primary) 16G85 (Secondary) 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. |
| title | Grothendieck Topologies and Sheaf Theory for Data and Graphs: An Approach Through Cech Closure Spaces |
| topic | Algebraic Topology 55N30, 55N31 (Primary) 16G85 (Secondary) |
| url | https://arxiv.org/abs/2109.13867 |