Left-exact localizations of ∞-topoi II: Grothendieck topologies

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.