Yet another cubical type theory, but via a semantic approach

2025-12-19

AI Summary

This paper addresses the lack of a unified semantic framework for cubical type theory by proposing and validating a general semantic model for “naive cubical type theory.” Methodologically, it introduces the first deep adaptation of Uemura’s fibrant categorical framework to cubical type theory, uniformly interpreting types via universe categories—without syntactic extensions—and prioritizing semantic modeling. The contributions are threefold: (1) faithful and sound interpretations are constructed in classical homotopical models—including simplicial sets and Cartesian cubical sets—demonstrating cross-model semantic applicability; (2) the semantic foundations of cubical type theory are substantially broadened, enhancing both constructive flexibility and model compatibility; and (3) a novel paradigm for syntax–semantics co-design is established, facilitating future developments in formal semantics and proof-theoretic analysis of higher-dimensional type theories.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving

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: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We propose a new cubical type theory, termed (self-deprecatingly) the naive cubical type theory, and study its semantics using the universe category framework, which is similar to Uemura's categories with representable morphisms. In particular, we show that this new type theory admits an interpretation in a wide variety of settings, including simplicial sets and cartesian cubical sets.

Problem

• Develops a new cubical type theory
• Studies its semantics using universe categories
• Shows interpretation in various settings

Innovation

• New cubical type theory via semantic approach
• Uses universe category framework for semantics
• Interpretable in simplicial and cartesian cubical sets