Left-exact Localizations of $\infty$-Topoi II: Grothendieck Topologies

TL;DR

This paper develops a comprehensive framework for extended Grothendieck topologies on $mbda$-topoi, establishing their equivalence with various localizations and topologies, and introduces tools like forcing for computations.

Contribution

It introduces extended Grothendieck topologies on $mbda$-topoi and proves their equivalence with multiple localization and topology notions, expanding the theoretical landscape.

Findings

Poset of extended Grothendieck topologies is a small frame.

Equivalence between extended Grothendieck topologies and various localizations.

Introduction of forcing as a computational tool for localizations.

Abstract

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.