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Functorial invariants for chaos topology from data

Denisse Sciamarella*

  • *Contact author: denisse.sciamarella@cnrs.fr
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Phys. Rev. E 114, 024205 – Published 10 August, 2026

DOI: https://doi.org/10.1103/zx76-5115

Abstract

The templex is a topological object bridging homologies and templates for chaotic dynamics. This article places the templex within category theory, introducing a directed path algebra, an edge operator on directed paths, and an equivalence relation for directed cycles that is distinct from directed homologies. The resulting functorial invariants are of two kinds: Abelian-group invariants, namely the homology groups, and semigroup invariants, namely the generatex semigroups. These invariants are separable through forgetful functors and constitute a robust framework for identifying tipping points, disambiguating physical mechanisms, and benchmarking data-driven models against observations or simulations. The formulation sets forth a nonmetric criterion for chaos from finite-time data and reveals that the concatenable nature of topological modes of variability is a direct consequence of the semigroup structure of the directed path algebra. Two applications are presented: an experimental speech signal and a climatic numerical simulation.

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Phys. Rev. E 114, 024205– Published 10 August, 2026
  • Received 22 October 2025
  • Accepted 23 July 2026
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