Filtered groupoid persistence: A homotopy-theoretic framework for symmetry-aware topological data analysis

Classical persistent homology extracts multiscale topological features from data but treats every point as an independent entity, making it insensitive to symmetries and equivalence relations intrinsic to physical, biological, and network datasets. We introduce filtered groupoid persistence (FGP), a mathematically rigorous framework that encodes both metric proximity and group-symmetry identifications within a single, unambiguous categorical structure: The action groupoid $ (\mathcal{G}_ \varepsilon)_{ \varepsilon\ge 0} $ of a finite group $ G $ acting simplicially on the Vietoris–Rips filtration $ (\mathrm{VR}_ \varepsilon(X))_{ \varepsilon\ge 0} $ of a finite metric space $ X $. The classifying space of this action groupoid is, by a standard identity, the Borel construction $ B \mathcal{G}_ \varepsilon \simeq EG\times_G|\mathrm{VR}_ \varepsilon(X)| $, and persistent homology of this filtered space defines the FGP persistence module $ M_k(\varepsilon) = H_k(B \mathcal{G}_ \varepsilon; \mathbb{F}) $. Grounding the construction in the classical action groupoid removes the ambiguity present in earlier drafts of this framework and lets us prove, rather than merely assert, that FGP specializes to classical persistent homology when $ G $ is trivial. We establish four principal results: An exact triviality proposition recovering classical persistent homology PH as the $ G = \{e\} $ case; a functoriality theorem showing that FGP is a well-defined functor from filtered $ G $-spaces to persistence modules; a symmetry-collapse theorem, proved via the standard fact that the Borel construction of a free $ G $-action is homotopy equivalent to the genuine quotient, characterizing precisely when group actions cause homological cycles to vanish; and a quantitative stability theorem, now proved directly for the orbit-quotient pseudometric space that the practical algorithm consumes, so that it holds unconditionally at every scale rather than only below a free threshold. We further prove that, when $ G $ acts freely below an explicit scale threshold, the practical algorithm (Vietoris-Rips persistence on an orbit-identified distance matrix) has a provable one-sided relationship to the categorical FGP module in degree one, as it can only under-report loops, never fabricate them. This closes part of the gap between the categorical definition and the computational pipeline while stating plainly, and corrects a previous overclaim, that the two do not coincide as simplicial complexes. The flag complex built from the quotient pseudometric can contain higher simplices with no consistent lift to the original space, so equality is not asserted in general and is left open in degree two and above. For actions that are not free (approximate or soft symmetries), we are explicit that the algorithm is a heuristic extension without a proven equivalence. We validate FGP on four synthetic datasets: A noisy circle under antipodal $ \mathbb{Z}_2 $-symmetry, a figure-eight under reflection symmetry, a flat torus under $ (\mathbb{Z}_2)^2 $-symmetry, and a Lorenz-attractor point cloud with approximate $ \mathbb{Z}_2 $-symmetry, reporting bottleneck distances, repeated-trial statistics, and the free-action scale threshold $ \varepsilon_{\mathrm{free}} $ itself alongside the qualitative comparison. In the two experiments where the relevant symmetry is genuinely free away from an easily avoided small locus (circle, figure-eight), FGP substantially reduces the spurious cycle count predicted by classical persistent homology, though we show that $ \varepsilon_{\mathrm{free}} $ is small enough in the figure-eight case that the observed dominant loop is only partially covered by the free-action guarantee. In the torus and Lorenz cases, where a fixed locus or only approximate symmetry is present, FGP reduces but does not eliminate spurious features, and we report these residuals honestly rather than as clean zeros.