Left-exact Localizations of $\infty$-Topoi I: Higher Sheaves

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