Left-exact Localizations of \(\infty\)-Topoi I: Higher Sheaves

We are developing tools for working with arbitrary left-exact localizations of \(\infty\)-topoi. We introduce a notion of higher sheaf with respect to an arbitrary set of maps \(\Sigma\) in an \(\infty\)-topos \(\mathscr{E}\). We show that the full subcategory of higher sheaves \(\mathrm{Sh}(\mathsc...

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Bibliographic Details
Published in:arXiv.org 2022-03
Main Authors: Anel, Mathieu, Biedermann, Georg, Finster, Eric, Joyal, André
Format: Article
Language:English
Subjects:
Factorization
Sheaves
Online Access:Get full text
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Summary:We are developing tools for working with arbitrary left-exact localizations of \(\infty\)-topoi. We introduce a notion of higher sheaf with respect to an arbitrary set of maps \(\Sigma\) in an \(\infty\)-topos \(\mathscr{E}\). We show that the full subcategory of higher sheaves \(\mathrm{Sh}(\mathscr{E},\Sigma)\) is an \(\infty\)-topos, and that the sheaf reflection \(\mathscr{E}\to \mathrm{Sh}(\mathscr{E},\Sigma)\) is the left-exact localization generated by \(\Sigma\). The proof depends on the notion of congruence, which is a substitute for the notion of Grothendieck topology in 1-topos theory.
ISSN:2331-8422
DOI:10.48550/arxiv.2101.02791