The Abhary-Grothendieck Topos: Detecting Spectral Anomalies via Cellular Sheaf Cohomology in Hyperspectral Remote Sensing

Abstract: Conventional hyperspectral remote sensing relies on Euclidean vector space models that fail to capture the local topological consistency of geological fabrics. In this paper, we introduce a paradigm shift by modeling the mineralized crust as a Grothendieck Topos. We construct a Cellular Sheaf ℱ over the simplicial complex of the topography, where stalks represent spectral bands and restriction maps encode lithological continuity. By computing the Sheaf Cohomology groups 𝐻 𝑘 (𝐾;ℱ), we demonstrate that spectral anomalies (ore bodies) manifest as non-trivial classes in the first cohomology group 𝐻 ۱, acting as obstructions to the global extension of local sections. We derive the Sheaf Laplacian operator to filter noise from signal and validate this approach using Sheaf Neural Networks (SNNs), demonstrating that the topological approach captures deep-seated mineralization patterns invisible to standard convolutional methods.