Abstract
We revisit the work of Toën–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 ∞-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 ∞-topoi. We use this in particular to show that the topological part of a left-exact localization of an ∞-topos is universally forcing the generators of this localization to be ∞-connected instead of inverting them.
We also introduce the notion of forcing, which is a tool to compute with localizations of ∞-topoi. We use this in particular to show that the topological part of a left-exact localization of an ∞-topos is universally forcing the generators of this localization to be ∞-connected instead of inverting them.
Original language | English |
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Article number | 107472 |
Number of pages | 63 |
Journal | Journal of Pure and Applied Algebra |
Volume | 228 |
Issue number | 3 |
Early online date | 12 Jun 2023 |
DOIs | |
Publication status | Published - Mar 2024 |
Bibliographical note
Acknowledgments:The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Mathieu Anel reports financial support was provided by Air Force Office of Scientific Research through grant FA9550-20-1-0305. Andre Joyal reports financial support was provided by Natural Sciences and Engineering Research Council of Canada through grant 371436.
Keywords
- ∞-topos
- Grothendieck topologies
- Hypercompletion
- Acyclic classes