HilbMult: A Banach-Enriched Multicategory for Operator Algebras
Shih-Yu Chang
TL;DR
This work builds a Banach-enriched symmetric monoidal multicategory framework, culminating in the HilbMult construction, to fuse operator theory with multicategory theory. It establishes a Functorial Spectral Theorem that extends the continuous functional calculus to a multimorphic setting, proves covariance under unitary transformations, and posits a universal target property for semantic representations of self-adjoint operators. By developing representability, adjoints, and a polynomial multicategory calculus, it provides a rigorous categorical semantics for operator calculus and spectral data. The proposed Categorical Spectral Architecture outlines a long-term program to harmonize operator theory, category theory, and noncommutative geometry, with practical implications for how multi-input operator networks are modeled and reasoned about. Overall, the paper provides both a concrete categorical infrastructure and a visionary program for unifying analytic and semantic dimensions of operator theory.
Abstract
Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle multi-input operations. These theories enable modular reasoning and coherent composition of complex systems, and have found applications in computer science, physics, and mathematics, including programming language semantics, quantum processes, tensor networks, operads, and higher algebra. Operator theory, in contrast, studies linear and multilinear transformations in functional spaces, forming the analytic backbone of modern analysis and quantum mechanics, with applications ranging from signal processing and control theory to data science. This paper explores the synergy between these two areas by showing how operator theory provides concrete analytic structures that naturally enrich multicategories, while multicategory theory supplies a unifying framework for organizing multi-input operators and ensuring coherence in complex networks. We develop a comprehensive categorical framework integrating operator theory with multicategories, introducing a symmetric monoidal multicategory of Hilbert spaces with bounded multilinear maps and establishing its enrichment and coherence properties. The framework includes a functorial spectral theorem, covariance under unitary transformations, and universality as a semantic target, providing a unified language for linking analytic structure, categorical semantics, and operator representations. This work lays the foundation for a broader research program uniting operator theory, category theory, and noncommutative geometry.