Univalence in locally cartesian closed infinity-categories
Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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
Originalsprog | Engelsk |
---|---|
Tidsskrift | Forum Mathematicum |
Vol/bind | 29 |
Udgave nummer | 3 |
Sider (fra-til) | 617-652 |
Antal sider | 36 |
ISSN | 0933-7741 |
DOI | |
Status | Udgivet - maj 2017 |
Eksternt udgivet | Ja |
ID: 331498718