Univalence in locally cartesian closed infinity-categories
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Univalence in locally cartesian closed infinity-categories. / Gepner, David; Kock, Joachim.
I: Forum Mathematicum, Bind 29, Nr. 3, 05.2017, s. 617-652.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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 -
ID: 331498718