Homotopical Foundations of Ternary Gamma Modules and Higher Structural Invariants

TL;DR

The paper develops a homotopical framework for ternary $\Gamma$-modules by proving $\mathcal{T}\text{-Mod}$ is Barr-exact and monoidal closed, and by transferring a Quillen model structure to the simplicial category $s(\mathcal{T}\text{-Mod})$. It establishes that the derived category $D(\mathcal{T}\text{-Mod})$ is $3$-angulated, with triadic quadrilaterals governed by the $\Gamma$-action, and defines non-abelian derived functors via simplicial resolutions. This yields a $3$-ary long exact sequence and $\Gamma$-linear connecting morphisms, providing a homological bridge to Nambu mechanics and $\mathbb{F}_1$-geometry. The work opens avenues for triadic arithmetic geometry, triadic motives, and higher-arity homological theories, suggesting a robust structural paradigm beyond binary homological algebra.

Abstract

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 $\mathbb{F}_1$.

Theorems & Definitions (15)

• Definition 1: Ternary $\Gamma$-Semiring
• Definition 2: Ternary $\Gamma$-Module
• Definition 3
• Proposition 1
• Definition 4: Ternary Tensor Product
• Theorem 1
• Theorem 2: Ternary Model Structure
• proof
• Theorem 3: Intrinsic 3-Angulation
• proof