Univalent categories and the Rezk completion

TL;DR

This paper develops a framework for univalent categories within Homotopy Type Theory, establishing a universal construction akin to the Rezk completion, which aligns equality with isomorphism in a homotopical setting.

Contribution

It introduces a definition of univalent categories in Univalent Foundations and shows how any category can be universally completed to a univalent one, connecting to Rezk completion.

Findings

Univalent categories satisfy the Univalence Axiom.

Any category can be weakly equivalently completed to a univalent category.

The construction relates to Rezk completion and stack completion in homotopical semantics.

Abstract

We develop category theory within Univalent Foundations, which is a foundational system for mathematics based on a homotopical interpretation of dependent type theory. In this system, we propose a definition of "category" for which equality and equivalence of categories agree. Such categories satisfy a version of the Univalence Axiom, saying that the type of isomorphisms between any two objects is equivalent to the identity type between these objects; we call them "saturated" or "univalent" categories. Moreover, we show that any category is weakly equivalent to a univalent one in a universal way. In homotopical and higher-categorical semantics, this construction corresponds to a truncated version of the Rezk completion for Segal spaces, and also to the stack completion of a prestack.