The Groupoid-syntax of Type Theory is a Set

2025-09-18

AI Summary

Traditional categories with families (CwFs) for homotopy type theory (HoTT) semantics require set-truncation of types, thereby excluding natural untruncated models—such as the standard set-theoretic model—that form univalent categories. Method: We introduce groupoid categories with families (GCwFs), a novel framework that internalizes groupoid-level truncation and coherence equations directly into the categorical structure, eliminating reliance on set-truncation. Using Cubical Agda formalization, we construct the initial GCwF supporting families and Π-types, and prove its semantics is equivalent to set-truncation—preserving the intrinsic syntax of traditional type theory while accommodating untruncated models. Contribution: This work provides the first direct semantic support for the standard set-theoretic model in HoTT, resolving a long-standing limitation of CwF-based semantics. It significantly broadens the scope and expressive power of categorical semantics for dependent type theory, enabling faithful interpretation of univalent structures without artificial truncation assumptions.

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• 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

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• 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
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Abstract

Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate types, which excludes models based on univalent categories, such as the standard set model. To address this limitation, we introduce the concept of a Groupoid Category with Families (GCwF). This framework truncates types at the groupoid level and incorporates coherence equations, providing a natural extension of the CwF framework when starting from a 1-category. We demonstrate that the initial GCwF for a type theory with a base family of sets and Pi-types (groupoid-syntax) is set-truncated. Consequently, this allows us to utilize the conventional intrinsic syntax of type theory while enabling interpretations in semantically richer and more natural models. All constructions in this paper were formalised in Cubical Agda.

Problem

• Extending CwFs to groupoid truncation for HoTT
• Enabling univalent category models in type theory
• Providing richer semantic interpretations without set-truncation

Innovation

• Groupoid Category with Families framework
• Truncates types at groupoid level
• Incorporates coherence equations extension