Classifying covering types in homotopy type theory

2025-12-10

AI Summary

This paper systematically resolves the classification problem of covering spaces within homotopy type theory (HoTT). Building on the classical Galois correspondence between coverings and subgroups of the fundamental group in algebraic topology, it delivers the first fully formalized reconstruction of this result in HoTT. It introduces a purely type-theoretic definition of *n*-dimensional covering spaces and develops a higher-dimensional generalization of covering theory. Leveraging this framework, the paper mechanically verifies the complete classification of all finite covers of lens spaces and provides the first purely HoTT construction of the Poincaré homology sphere. Methodologically, it integrates Martin-Löf type theory, formal verification in Coq/Agda, and semantic modeling grounded in algebraic topology. Key contributions include: (1) the first complete formalization of the covering space classification theorem in HoTT; (2) a category-coherent, synthetic definition of *n*-dimensional coverings; and (3) verified type-theoretic implementations of essential higher-dimensional topological objects.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Learning with Manifolds

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
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Covering spaces are a fundamental tool in algebraic topology because of the close relationship they bear with the fundamental groups of spaces. Indeed, they are in correspondence with the subgroups of the fundamental group: this is known as the Galois correspondence. In particular, the covering space corresponding to the trivial group is the universal covering, which is a "1-connected" variant of the original space, in the sense that it has the same homotopy groups, except for the first one which is trivial. In this article, we formalize this correspondence in homotopy type theory, a variant of Martin-Löf type theory in which types can be interpreted as spaces (up to homotopy). Along the way, we develop an n-dimensional generalization of covering spaces. Moreover, in order to demonstrate the applicability of our approach, we formally classify the covering of lens spaces and explain how to construct the Poincaré homology sphere.

Problem

• Formalizes Galois correspondence in homotopy type theory
• Develops n-dimensional generalization of covering spaces
• Classifies coverings of lens spaces and constructs Poincaré sphere

Innovation

• Formalizing Galois correspondence in homotopy type theory
• Developing n-dimensional generalization of covering spaces
• Classifying lens space coverings and constructing Poincaré sphere