HilbMult: A Banach-Enriched Multicategory for Operator Algebras
Shih-Yu Chang

TL;DR
This paper develops a categorical framework that integrates operator theory with multicategory theory, providing a unified language for multi-input operators in functional analysis and quantum mechanics.
Contribution
It introduces a Banach-enriched multicategory of Hilbert spaces with bounded multilinear maps, establishing coherence, enrichment, and a functorial spectral theorem.
Findings
Established a symmetric monoidal multicategory of Hilbert spaces
Proved enrichment and coherence properties of the framework
Developed a functorial spectral theorem for the structure
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…
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Taxonomy
TopicsLogic, programming, and type systems · Formal Methods in Verification · Algebraic and Geometric Analysis
