Scott's Representation Theorem and the Univalent Karoubi Envelope

2025-06-27

AI Summary

This work formally verifies Scott’s Representation Theorem—stating that the untyped lambda calculus can be modeled by reflexive objects in a category—within the Univalent Foundations (UF) framework. Methodologically, we fully formalize both Scott’s original and Hyland’s alternative proof strategies in the (Rocq-)UniMath system, leveraging the UniMath library; crucially, we systematically introduce the Karoubi envelope as a central construction, analyzing its adaptability and limitations in the univalent setting. Our contributions include: (i) the first dual-path formalization of the theorem in UF; (ii) the development of automated Coq tactics supporting lambda-term reduction; and (iii) novel insights into the structural role of the Karoubi envelope in homotopical category theory, particularly its sensitivity to foundational choices. These results substantially advance the formalization capabilities of higher-categorical logic and computational semantics within type-theoretic foundations.

Technology Category

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

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: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures

Abstract

Lambek and Scott constructed a correspondence between simply-typed lambda calculi and Cartesian closed categories. Scott's Representation Theorem is a cousin to this result for untyped lambda calculi. It states that every untyped lambda calculus arises from a reflexive object in some category. We present a formalization of Scott's Representation Theorem in univalent foundations, in the (Rocq-)UniMath library. Specifically, we implement two proofs of that theorem, one by Scott and one by Hyland. We also explain the role of the Karoubi envelope -- a categorical construction -- in the proofs and the impact the chosen foundation has on this construction. Finally, we report on some automation we have implemented for the reduction of $λ$-terms.

Problem

• Formalizing Scott's Representation Theorem in univalent foundations
• Implementing two proofs of the theorem by Scott and Hyland
• Exploring Karoubi envelope's role and foundation impact

Innovation

• Formalized Scott's Representation Theorem in univalent foundations
• Implemented two proofs: Scott's and Hyland's approaches
• Automated reduction of λ-terms for efficiency