Higher sheaves and left-exact localizations of ∞-topoi

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal

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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.
Original languageEnglish
Article number108268
Number of pages64
JournalAdvances in Mathematics
Early online date28 Feb 2022
Publication statusPublished - 14 May 2022


  • Infinity-topos
  • Left-exact localization
  • Sheaf
  • Site
  • Congruence
  • Acyclic class


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