TY - JOUR

T1 - Univalence in locally cartesian closed infinity-categories

AU - Gepner, David

AU - Kock, Joachim

PY - 2017/5

Y1 - 2017/5

N2 - After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-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 infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (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 infinity-quasitopoi (certain infinity-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) truncated, as well as some univalent families in the Morel-Voevodsky infinity-category of motivic spaces, an instance of a locally cartesian closed infinity-category which is not an n-topos for any 0

AB - After developing the basic theory of locally cartesian localizations of presentable locally cartesian closed infinity-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 infinity-topos has a hierarchy of "universal" univalent families, indexed by regular cardinals, and that n-topoi have univalent families classifying (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 infinity-quasitopoi (certain infinity-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) truncated, as well as some univalent families in the Morel-Voevodsky infinity-category of motivic spaces, an instance of a locally cartesian closed infinity-category which is not an n-topos for any 0

KW - Univalence

KW - infinity-categories

KW - infinity-topoi

KW - infinity-quasitopoi

KW - factorization systems

KW - localization

KW - FACTORIZATION SYSTEMS

KW - MODEL

U2 - 10.1515/forum-2015-0228

DO - 10.1515/forum-2015-0228

M3 - Journal article

VL - 29

SP - 617

EP - 652

JO - Forum Mathematicum

JF - Forum Mathematicum

SN - 0933-7741

IS - 3

ER -