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• W2556331506 abstract "Abstract After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -topos has a hierarchy of “universal” univalent families, indexed by regular cardinals, and that n -topoi have univalent families classifying <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(n-2)}$ -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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> $infty$ -quasitopoi (certain <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -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 <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> ${(n-2)}$ -truncated, as well as some univalent families in the Morel–Voevodsky <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -category of motivic spaces, an instance of a locally cartesian closed <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -category which is not an n -topos for any <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>n</m:mi> <m:mo>≤</m:mo> <m:mi>∞</m:mi> </m:mrow> </m:math> ${0leq nleqinfty}$ . Lastly, we show that any presentable locally cartesian closed <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -category is modeled by a combinatorial type-theoretic model category, and conversely that the <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -category underlying a combinatorial type-theoretic model category is presentable and locally cartesian closed. Under this correspondence, univalent families in presentable locally cartesian closed <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>∞</m:mi> </m:math> ${infty}$ -categories correspond to univalent fibrations in combinatorial type-theoretic model categories." @default.
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• W2556331506 date "2016-11-06" @default.
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• W2556331506 title "Univalence in locally cartesian closed ∞-categories" @default.
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• W2556331506 doi "https://doi.org/10.1515/forum-2015-0228" @default.
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