Left-exact Localizations of $\infty$-Topoi II: Grothendieck Topologies
| Autor: | Anel, Mathieu, Biedermann, Georg, Finster, Eric, Joyal, André |
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| Rok vydání: | 2022 |
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| Druh dokumentu: | Working Paper |
| Popis: | We revisit the work of To\"en--Vezzosi and Lurie on Grothendieck topologies, using the new tools of acyclic classes and congruences. We introduce a notion of extended Grothendieck topology on any $\infty$-topos, and prove that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere--Tierney topologies, and covering topologies (a variation on the notion of pretopology). It follows that these posets are small and have the structure of a frame. We revisit also the topological--cotopological factorization by introducing the notion of a cotopological morphism. And we revisit the notions of hypercompletion, hyperdescent, hypercoverings and hypersheaves associated to an extended Grothendieck topology. We also introduce the notion of forcing, which is a tool to compute with localizations of $\infty$-topoi. Comment: v2. To be published in JPAA. We change some terminology and notations. We simplified the proof of Proposition 3.3.6 and reorganized Section 2 around that |
| Databáze: | arXiv |
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