# Prime Coordinate Functor A fundamental functor φ between the category of objects and the category of graded modules, transforming multiplicative structures into additive ones. ## Definition The prime-coordinate map can be understood as a fundamental functor between categories: - Base Categories: Define Obj as the category of objects with composition and Mod as the category of graded modules. - The Prime Functor: The map φ: Obj → Mod is a functor satisfying: φ(X ⊗ Y) = φ(X) ⊕ φ(Y) and φ(1_Obj) = 0_Mod, where ⊗ is composition in Obj and ⊕ is addition in Mod. - Natural Transformation: The homomorphism property of φ manifests as a natural transformation η: ⊗ ⟹ φ^*⊕ between the composition functor and the pulled-back addition functor. - Contravariant Properties: For inverse operations, φ exhibits contravariance: φ(X^{-1}) = -φ(X), ensuring compatibility with the group structure in both domains. ## Mathematical Formulation $ \phi: Obj \to Mod \text{ is a functor such that:} $ $ \phi(X \otimes Y) = \phi(X) \oplus \phi(Y) $ $ \phi(1_{Obj}) = 0_{Mod} $ $ \text{For inverse operations: } \phi(X^{-1}) = -\phi(X) $ ## Related Concepts - [[uor-c-004|canonical-representation]] - [[uor-c-002|prime-decomposition]] ## Metadata - **ID:** urn:uor:concept:prime-coordinate-functor - **Code:** UOR-C-027