Univalent categories and the Rezk completion
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.
Cite
@article{arxiv.1303.0584,
title = {Univalent categories and the Rezk completion},
author = {Benedikt Ahrens and Chris Kapulkin and Michael Shulman},
journal= {arXiv preprint arXiv:1303.0584},
year = {2019}
}
Comments
27 pages, ancillary files contain formalized proofs in the proof assistant Coq; v2: version for publication in Math. Struct. in Comp. Sci., incorporating suggestions by referees and Voevodsky