Abstract
We are developing tools for working with arbitrary left-exact localizations of ∞-topoi. We introduce a notion of higher sheaf with respect to an arbitrary set of maps Σ in an ∞-topos E. We show that the full subcategory of higher sheaves Sh(E,Σ) is an ∞-topos, and that the sheaf reflection E→Sh(E,Σ) is the left-exact localization generated by Σ. The proof depends on the notion of congruence, which is a substitute for the notion of Grothendieck topology in 1-topos theory.
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