String Diagrams for Monoidal Categories, in Rocq

2026-02-23

AI Summary

This work addresses the challenge that equational reasoning in monoidal categories is often cumbersome and lacks intuitiveness, with existing tools struggling to combine graphical manipulation and formal verification. To bridge this gap, the paper presents a novel library implemented in Rocq for monoidal categories, which seamlessly integrates graphical rewriting via string diagrams with the generation of formally verified proofs. By automating the derivation of MacLane coherence isomorphisms, the system enables users to reason with rigorous categorical semantics while interfacing with external visualization tools. This integration supports an automatic translation from graphical operations into concise, human-readable, and machine-verifiable textual proofs, thereby significantly enhancing the intuitiveness, reliability, and degree of automation in reasoning within monoidal categories.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Computer Vision: Visual Reasoning & Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We present a Rocq library for monoidal categories, which includes a decision procedure for proving equality of morphisms as well as notations that make it possible to reason as if they were strict, inferring MacLane isomorphims automatically in the background. Together with an external tool for visualising and editing string diagrams, this make it possible to perform rewriting steps in monoidal categories graphically, and to translate them into textual formal proofs which are concise and readable.

Problem

• monoidal categories
• string diagrams
• morphism equality
• formal proof
• graphical reasoning

Innovation

• monoidal categories
• string diagrams
• decision procedure
• MacLane coherence
• formal proof