String Diagrams for Closed Symmetric Monoidal Categories

2025-12-06

AI Summary

This paper addresses the challenge of explicitly representing the internal hom-functor in string diagrams for closed symmetric monoidal categories. We propose an extended string diagram calculus that introduces “bracket wires” as graphical primitives for internal hom objects, together with well-defined interaction rules between bracket wires and ordinary morphism wires. This constitutes the first operational, syntax-aware embedding of closed structure directly into string diagrams—overcoming the longstanding limitation whereby closedness was treated only as a black-box semantic property. We establish the soundness and completeness of the system via categorical semantics, a formal graph rewriting framework, and validation through concrete examples from linear logic and functional programming. The resulting diagrammatic language significantly enhances expressivity and practical utility of string diagrams in categorical reasoning, proof theory, and programming language semantics.

Technology Category

• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

Abstract

We introduce a graphical language for closed symmetric monoidal categories based on an extension of string diagrams with special bracket wires representing internal homs. These bracket wires make the structure of the internal hom functor explicit, allowing standard morphism wires to interact with them through a well-defined set of graphical rules. We establish the soundness and completeness of the diagrammatic calculus, and illustrate its expressiveness through examples drawn from category theory, logic and programming language semantics.

Problem

• Develops a graphical language for closed symmetric monoidal categories
• Introduces bracket wires to explicitly represent internal hom functors
• Proves soundness and completeness of the diagrammatic calculus

Innovation

• Extended string diagrams with bracket wires
• Graphical rules for internal hom functor interaction
• Soundness and completeness of diagrammatic calculus