Functorial invariants for chaos topology from data
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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BibTeX
@article{sciamarella2026functorial,
title = {{Functorial invariants for chaos topology from data}},
author = {Denisse Sciamarella},
journal = {Physical Review E},
year = {2026},
month = aug,
doi = {10.1103/zx76-5115},
eprint = {2602.18463},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05759679v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Functorial invariants for chaos topology from data
AU - Denisse Sciamarella
PY - 2026
DA - 2026/08/10
JO - Physical Review E
DO - 10.1103/zx76-5115
UR - https://hal.science/hal-05759679v1
AB - 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.
SN - 2470-0045
SN - 2470-0053
ER -