2026/02/06 by Anonymous, Denisse Sciamarella
Computer Science · Mathematics · #Algebra over a field #Equivalence (formal languages) #Equivalence relation #Functor #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Mathematical Dynamics and Fractals #Path (computing) #Semigroup #Separable space #Topological and Geometric Data Analysis #Topology (electrical circuits)
paper · pdf · doi:10.1103/zx76-5115
published in Physical review. E (American Physical Society)
openalex publication_date 2026/07/23 · openalex created_date 2026/07/24 · openalex updated_date 2026/08/01
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.