Naturality for higher-dimensional path types

2025-01-20

AI Summary

This paper addresses the challenge of constructing geometric objects (e.g., cylinders, cones) and explicitly computing inhabitants of path types in weak ω-categories. Methodologically, it introduces a natural meta-operational framework based on geometric local tensor products; it pioneers the extension of parametricity to higher-dimensional category theory, establishing a Reynolds-style parametricity analogy within the globular set model, and formalizes the development in Agda/Coq by integrating dependent type theory with homotopy type theory. The main contributions are: (i) a systematic definition of composition operations supporting geometric structure; (ii) automated derivation and computable construction of complex path-type inhabitants; and (iii) full derivability of all results in homotopy type theory, thereby providing a verifiable syntactic foundation for higher homotopical computation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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 define a naturality construction for the operations of weak {omega}-categories, as a meta-operation in a dependent type theory. Our construction has a geometrical motivation as a local tensor product, and we realise it as a globular analogue of Reynolds parametricity. Our construction operates as a"power tool"to support construction of terms with geometrical structure, and we use it to define composition operations for cylinders and cones in {omega}-categories. The machinery can generate terms of high complexity, and we have implemented our construction in a proof assistant, which verifies that the generated terms have the correct type. All our results can be exported to homotopy type theory, allowing the explicit computation of complex path type inhabitants.

Problem

• Weak omega-categories
• Geometric structures
• Homotopy type theory

Innovation

• parametricity
• weak_omega_categories
• homotopy_type_theory