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arXiv · 2602.18463

Functorial invariants for chaos topology from data

Denisse Sciamarella
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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 non-metric 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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Denisse Sciamarella. 2026-02-06. Functorial invariants for chaos topology from data. https://doi.org/10.1103/zx76-5115

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