  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.
