Sheaf-Theoretic Persistent Cohomology over Action Groupoids: A Sym-metry-Aware Topological Data Analysis Framework for Computational and Bio-medical Data

Abstract

Classical persistent homology summarizes a dataset's shape from metric filtrations alone, treating every data point as algebraically and symmetrically interchangeable. This is a poor fit for computational and biomedical data that carries known group symmetry because symmetry-related copies of a feature are counted as if they were topologically distinct, inflating Betti numbers and degrading downstream classification and clustering pipelines. We introduce Sheaf-Theoretic Homotopy Groupoid Persistence (S-HGP), a framework that computes persistent cohomology of the classifying space of a data-driven action groupoid, optionally with sheaf (local-system) coefficients that encode orientation, monodromy, or other locally varying algebraic structure. Classical persistent cohomology is recovered as the special case of a constant sheaf on a trivial groupoid. We prove functoriality and Gromov-Hausdorff stability of the resulting persistence modules, a Mayer–Vietoris spectral sequence for local-to-global computation, and a symmetry-collapse theorem describing when S-HGP provably returns fewer, correct features than classical persistent homology. We validate the framework with fully reproducible numerical experiments: (i) a corrected torus baseline recovering the true Betti number; (ii) a parametrized family of symmetry-replicated synthetic datasets (, ,  group actions) in which classical persistent homology provably overcounts loops in exact proportion to the group order while the groupoid-quotient computation recovers the correct rank in every case; (iii) a direct evaluation of the sheaf cochain complex on constant versus orientation-twisted coefficients over a discretized circle, confirming the theoretically predicted collapse of cohomology under a nontrivial local system; (iv) a synthetic homomeric protein-assembly point cloud with discrete rotational symmetry, on which the groupoid quotient recovers exactly one significant topological feature per protomer while correctly retaining a genuine assembly-level macro-scale feature; and (v) a synthetic bilateral connectome-like network, on which classical persistent homology exhibits the predicted near-doubling of topological features that the groupoid quotient removes without discarding genuine inter-hemispheric structure. We discuss applications to equivariant machine learning, protein structure analysis, and brain network connectomics, with two given a first synthetic numerical demonstration; all code, seeds, and raw output are reported for full reproducibility.