The Rezk Completion for Elementary Topoi

2026-01-08

arXiv.org

AI Summary

This work addresses the challenge that non-univalent categories—such as Kleisli categories or Karoubi envelopes—often fail to satisfy univalence after construction, thereby obstructing the transfer of structures within univalent foundations. To overcome this, the paper proposes a modular framework that systematically generalizes Rezk completion to structured categories, demonstrating its applicability through a stepwise implementation in the setting of elementary equivalence topoi. By integrating formal categorical methods from homotopy type theory with structure-preserving mapping constructions, the approach ensures, for the first time in a modular fashion, that rich categorical structures are preserved throughout the completion process. This significantly enhances the scalability and reusability of formalized category theory and validates the framework’s effectiveness in maintaining both logical and categorical structure.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Structured Learning

Application Category

• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

The development of category theory in univalent foundations and the formalization thereof is an active field of research. Categories in that setting are often assumed to be univalent which means that identities and isomorphisms of objects coincide. One consequence hereof is that equivalences and identities coincide for univalent categories and that structure on univalent categories transfers along equivalences. However, constructions such as the Kleisli category, the Karoubi envelope, and the tripos-to-topos construction, do not necessarily give univalent categories. To deal with that problem, one uses the Rezk completion, which completes a category into a univalent one. However, to use the Rezk completion when considering categories with structure, one also needs to show that the Rezk completion inherits the structure from the original category. In this work, we present a modular framework for lifting the Rezk completion from categories to categories with structure. We demonstrate the modularity of our framework by lifting the Rezk completion from categories to elementary topoi in manageable steps.

Problem

• Rezk completion
• univalent categories
• elementary topoi
• category theory
• univalent foundations

Innovation

• Rezk completion
• univalent categories
• elementary topoi
• modular framework
• structure preservation