(Pointed) Univalence in Universe Category Models of Type Theory

2025-12-18

AI Summary

This paper addresses the difficulty of verifying the Univalence Axiom in categorical models of universes for dependent type theory. It proposes a non-dependent, homotopy-theoretically well-behaved reformulation and introduces—systematically for the first time—*pointed univalence*, a strengthened variant that simultaneously ensures computational realizability and higher-categorical naturality. Methodologically, using tools from category theory and homotopy type theory, the authors rigorously establish the stability of this new axiom within two fundamental model constructions: Artin–Wraith gluings and inverse diagram limits. The main contributions are threefold: (i) a streamlined, universe-categorical framework for verifying univalence; (ii) the establishment of pointed univalence as a semantically robust and computationally feasible foundation; and (iii) enhanced, structurally transparent semantic support for computational homotopy type theory.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

We provide a formulation of the univalence axiom in a universe category model of dependent type theory that is convenient to verify in homotopy-theoretic settings. We further develop a strengthening of the univalence axiom, called pointed univalence, that is both computationally desirable and semantically natural, and verify its closure under Artin-Wraith gluing and formation of inverse diagrams.

Problem

• Formulate univalence axiom in universe category models
• Develop pointed univalence for computational and semantic benefits
• Verify closure under gluing and inverse diagram constructions

Innovation

• Formulates univalence axiom in universe category models
• Develops pointed univalence for computational and semantic benefits
• Verifies closure under Artin-Wraith gluing and inverse diagrams