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.

arXiv commentFinal journal version for JPAA. Improved exposition (particularly Sections 2 and 5) and corrected minor mistakes in virtue of the comments of an anonymous referee

References in corpus (8)

• Higher Topos Theory
• Univalent categories and the Rezk completion
• The local universes model: an overlooked coherence construction for dependent type theories
• Univalence for inverse diagrams and homotopy canonicity
• Comparison of models for $(\infty ,n)$-categories, II
• Homotopy Theoretic Models of Type Theory
• Complete Segal Objects
• Locally Cartesian Closed Quasicategories from Type Theory

Cited by in corpus (1)

• Univalence in Higher Category Theory