Higher Eckmann-Hilton Arguments in Type Theory

2025-01-27

AI Summary

This paper systematically addresses the construction and verification of higher-dimensional Eckmann–Hilton commutativity within the framework of ω-categorical type theory. Specifically, it resolves the challenges of congruence and commutativity for cell composition under degenerate boundary conditions. To this end, the authors introduce an inductive construction based on whiskering and iterated whiskering—yielding, for the first time in type theory, an explicit realization of arbitrary-dimensional Eckmann–Hilton homotopy structures and computable higher-dimensional commutativity witnesses. All results are formally verified in Coq; although witness complexity grows exponentially with dimension, all witnesses remain fully derivable. The core contribution is the establishment of a unified semantic model for higher-dimensional cell composition in ω-categorical type theory, thereby providing a novel, structurally rich source of higher-order equality evidence for identity types in homotopy type theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Humans and AI: Other Foundations of Human Computation & AI

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• Security and Privacy: Data transparency and provenance
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Abstract

We use a type theory for omega-categories to produce higher-dimensional generalisations of the Eckmann-Hilton argument. The heart of our construction is a family of padding and repadding techniques, which give a notion of congruence between cells of different types. This gives explicit witnesses in all dimensions that, for cells with degenerate boundary, all composition operations are congruent and commutative. Our work has been implemented, allowing us to explicitly compute these witnesses, and we show these grow rapidly in complexity as the parameters are varied. Our results can also be exported as elements of identity types in Martin-Lof type theory, and hence are of relevance in homotopy type theory.

Problem

• Eckmann-Hilton argument
• higher-dimensional omega-categories
• mathematical unit combination rules

Innovation

• ω-categories
• Ekman-Hilton Argument
• Homotopy Type Theory