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 $\mathcal{E}$. We show that the full subcategory of higher sheaves $\mathrm{Sh}(\mathcal{E},\Sigma )$ is an $\infty$-topos, and that the sheaf reflection $\mathcal{E}\to \mathrm{Sh}(\mathcal{E},\Sigma )$ 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.

v2: 47 pages, paper substantially rewritten, proofs of main theorems shortened, examples added. v3: changed title, corrected a few typos, added Remark 3.3.10, split old Prop. 4.3.6 into new Prop. 4.3.6 and 4.3.7 for reference purposes. v4: published version, corrected typos and bibliography

References in corpus (1)

• Small object arguments, plus-construction, and left-exact localizations

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