Higher sheaves and left-exact localizations of ∞-topoi

Mathieu Anel; Georg Biedermann; Eric Finster; André Joyal

doi:10.1016/j.aim.2022.108268

Higher sheaves and left-exact localizations of ∞-topoi

Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal

Computer Science

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Abstract

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 language	English
Article number	108268
Number of pages	64
Journal	Advances in Mathematics
Volume	400
Early online date	28 Feb 2022
DOIs	https://doi.org/10.1016/j.aim.2022.108268
Publication status	Published - 14 May 2022

Keywords

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

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10.1016/j.aim.2022.108268

AnelM2022LeftAccepted author manuscript, 1.45 MBLicence: Creative Commons: Attribution-NonCommercial-NoDerivs (CC BY-NC-ND)

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abstract = "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 퓔. We show that the full subcategory of higher sheaves Sh (퓔, Σ) is an ∞-topos, and that the sheaf reflection 퓔→Sh (퓔, 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.",

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AU - Anel, Mathieu

AU - Biedermann, Georg

AU - Finster, Eric

AU - Joyal, André

PY - 2022/5/14

Y1 - 2022/5/14

N2 - 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 퓔. We show that the full subcategory of higher sheaves Sh (퓔, Σ) is an ∞-topos, and that the sheaf reflection 퓔→Sh (퓔, 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.

AB - 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 퓔. We show that the full subcategory of higher sheaves Sh (퓔, Σ) is an ∞-topos, and that the sheaf reflection 퓔→Sh (퓔, 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.

KW - Infinity-topos

KW - Left-exact localization

KW - Sheaf

KW - Site

KW - Congruence

KW - Acyclic class

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M3 - Article

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VL - 400

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 108268

ER -

Higher sheaves and left-exact localizations of ∞-topoi

Abstract

Keywords

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