A Unified Mathematical Framework for Distributed Data Fabrics: Categorical Hypergraph Models

Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous data management in distributed systems through a hypergraph-based structure $\mathcal{F} = (D, M, G, T, P, A)$. Datasets, metadata, transformations, policies, and analytics are modeled over a distributed system $\Sigma = (N, C)$, with multi-way relationships encoded in a hypergraph $G = (V, E)$. A categorical approach, with datasets as objects and transformations as morphisms, supports operations like data integration and federated learning. The hypergraph is embedded into a modular tensor category, capturing relational symmetries via braided monoidal structures, with geometric analogies to Hurwitz spaces enriching the algebraic modeling. We prove the NP-hardness of critical tasks, such as schema matching and dynamic partitioning, and propose spectral methods and symmetry-based alignments for scalable solutions. The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.