Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908763287453696 |
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| author | Gokavarapu, Chandrasekhar |
| author_facet | Gokavarapu, Chandrasekhar |
| contents | We establish a foundational homotopical framework for ternary $Γ$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary algebra by constructing a cofibrantly generated Quillen model structure on the simplicial category $s(\mathcal{T}\text{-Mod})$. Our central discovery is that the derived category $D(\mathcal{T}\text{-Mod})$ constitutes a 3-angulated category, where the derived periodicity is governed by triadic quadrilaterals rather than binary triangles. We derive the 3-ary long exact sequence and characterize the connecting morphisms as invariants of the $Γ$-parameter space. This framework provides a rigorous homological bridge to Nambu mechanics and absolute geometry over $\mathbb{F}_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08918 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants Gokavarapu, Chandrasekhar Rings and Algebras Primary 18G55, 18G15, Secondary 18E10, 16Y80, 14G40 We establish a foundational homotopical framework for ternary $Γ$-modules by establishing that $\mathcal{T}\text{-Mod}$ is a Barr-exact, monoidal closed category. We resolve the long-standing "additivity obstruction" in non-binary algebra by constructing a cofibrantly generated Quillen model structure on the simplicial category $s(\mathcal{T}\text{-Mod})$. Our central discovery is that the derived category $D(\mathcal{T}\text{-Mod})$ constitutes a 3-angulated category, where the derived periodicity is governed by triadic quadrilaterals rather than binary triangles. We derive the 3-ary long exact sequence and characterize the connecting morphisms as invariants of the $Γ$-parameter space. This framework provides a rigorous homological bridge to Nambu mechanics and absolute geometry over $\mathbb{F}_1$. |
| title | Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants |
| topic | Rings and Algebras Primary 18G55, 18G15, Secondary 18E10, 16Y80, 14G40 |
| url | https://arxiv.org/abs/2601.08918 |