Univalence in locally cartesian closed ∞-categories

Abstract

Abstract After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed ∞ ${\infty}$ -categories, we establish the representability of equivalences and show that univalent families, in the sense of Voevodsky, form a poset isomorphic to the poset of bounded local classes, in the sense of Lurie. It follows that every ∞ ${\infty}$ -topos has a hierarchy of “universal” univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying ( n - 2 ) ${(n-2)}$ -truncated maps. We show that univalent families are preserved (and detected) by right adjoints to locally cartesian localizations, and use this to exhibit certain canonical univalent families in ∞ $\infty$ -quasitopoi (certain ∞ ${\infty}$ -categories of “separated presheaves”, introduced here). We also exhibit some more exotic examples of univalent families, illustrating that a univalent family in an n-topos need not be ( n - 2 ) ${(n-2)}$ -truncated, as well as some univalent families in the Morel–Voevodsky ∞ ${\infty}$ -category of motivic spaces, an instance of a locally cartesian closed ∞ ${\infty}$ -category which is not an n-topos for any 0 ≤ n ≤ ∞ ${0\leq n\leq\infty}$ . Lastly, we show that any presentable locally cartesian closed ∞ ${\infty}$ -category is modeled by a combinatorial type-theoretic model category, and conversely that the ∞ ${\infty}$ -category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed ∞ ${\infty}$ -categories correspond to univalent fibrations in combinatorial type-theoretic model categories.