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Left-exact localizations of ∞-topoi I: Higher sheaves

2022; Elsevier BV; Volume: 400; Linguagem: Inglês

10.1016/j.aim.2022.108268

1090-2082

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal,

Algebraic structures and combinatorial models

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.