Univalence in locally cartesian closed infinity-categories
Research output: Contribution to journal › Journal article › Research › peer-review
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
Original language | English |
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Journal | Forum Mathematicum |
Volume | 29 |
Issue number | 3 |
Pages (from-to) | 617-652 |
Number of pages | 36 |
ISSN | 0933-7741 |
DOIs | |
Publication status | Published - May 2017 |
Externally published | Yes |
- Univalence, infinity-categories, infinity-topoi, infinity-quasitopoi, factorization systems, localization, FACTORIZATION SYSTEMS, MODEL
Research areas
ID: 331498718