{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T08:35:37Z","timestamp":1790930137048,"version":"4.1.0"},"reference-count":45,"publisher":"American Physical Society (APS)","issue":"2","license":[{"start":{"date-parts":[[2026,8,10]],"date-time":"2026-08-10T00:00:00Z","timestamp":1786320000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2026,7,23]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1103\/zx76-5115","type":"journal-article","created":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T14:27:37Z","timestamp":1784816857000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":1,"title":["Functorial invariants for chaos topology from data"],"prefix":"10.1103","volume":"114","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5218-3708","authenticated-orcid":true,"given":"Denisse","family":"Sciamarella","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/02feahw73","id-type":"ROR","asserted-by":"publisher"}],"name":"CNRS\u2013Centre National de la Recherche Scientifique"},{"id":[{"id":"https:\/\/ror.org\/03cqe8w59","id-type":"ROR","asserted-by":"publisher"}],"name":"CONICET"},{"id":[{"id":"https:\/\/ror.org\/05k07f122","id-type":"ROR","asserted-by":"publisher"}],"name":"Institut Franco-Argentin d'\u00c9tudes sur le Climat et ses Impacts"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,8,10]]},"reference":[{"key":"zx76-5115Cc1R1","doi-asserted-by":"publisher","DOI":"10.1146\/annurev-fluid-010719-060214","type":"journal-article"},{"key":"zx76-5115Cc2R1","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.70.1455","type":"journal-article"},{"key":"zx76-5115Cc3R1","doi-asserted-by":"publisher","DOI":"10.5194\/npg-30-399-2023","type":"journal-article"},{"key":"zx76-5115Cc4R1","doi-asserted-by":"publisher","DOI":"10.1142\/S0218127410027854","type":"journal-article"},{"key":"zx76-5115Cc5R1","volume-title":"The Topology of Chaos: Alice in Stretch and Squeezeland","author":"R. 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All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"024205"}}