The Rezk Completion for Elementary Topoi

TL;DR

This paper develops a modular framework to extend the Rezk completion process from basic categories to structured categories like elementary topoi within univalent foundations, ensuring structure inheritance.

Contribution

It introduces a modular approach for lifting the Rezk completion to categories with structure, demonstrated on elementary topoi.

Findings

Framework successfully lifts Rezk completion to elementary topoi.

Ensures structure inheritance during the Rezk completion process.

Facilitates formalization of structured categories in univalent foundations.

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.