Univalence and completeness of Segal objects

Abstract

Univalence, originally a type theoretical notion at the heart of Voevodsky's Univalent Foundations Program, has found general importance as a higher categorical property that characterizes descent and hence classifying maps in $(\infty,1)$-categories. Completeness is a property of Segal spaces introduced by Rezk that characterizes those Segal spaces which are $(\infty,1)$-categories. In this paper, first, we make rigorous an analogy between univalence and completeness that has found various informal expressions in the higher categorical research community to date, and second, study its ramifications. The core aspect of this analogy can be understood as a translation between internal and external notions, motivated by model categorical considerations of Joyal and Tierney. As a result, we characterize the internal notion of univalence in logical model categories by the external notion of completeness defined as the right Quillen condition of suitably indexed Set-weighted limit functors. Furthermore, we extend the analogy and show that univalent completion in the sense of van den Berg and Moerdijk translates to Rezk-completion of associated Segal objects as well. Motivated by these correspondences, we exhibit univalence as a homotopical locality condition whenever univalent completion exists.