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.
- Left-exact localization
- Acyclic class