Descent along atlases is equivalent to hyperdescent

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Let TT be an ∞\infty-category with a Grothendieck topology, let XX be an object of TT, and let F:Top→\SpaceF:T^{\mathrm{op}}\rightarrow\Space be an ∞\infty-presheaf. An atlas for an object V⊆XV\subseteq X is an ∞\infty-functor U:I→T/VU:I\rightarrow T_{/V} satisfying the covering condition specified in the paper, and FF satisfies descent along atlases if every atlas (I,U)(I,U) of every open V⊆XV\subseteq X induces an equivalence

F(V)≃lim⁡i:IF(Ui).F(V)\simeq\lim_{i:I}F(U_i).

Atlas descent conjecture. The following conditions on FF are equivalent: (1) FF satisfies descent along atlases; (2) FF satisfies hyperdescent. In particular, the full subcategory of Presheaf⁡∞(X)\operatorname{Presheaf}_\infty(X) spanned by functors satisfying descent along atlases is a hypercomplete topos.

The claim proposes that atlases provide an alternative formulation of hyperdescent, retaining the relevant descent theory while avoiding the potentially non-finite or less intuitive constructions involved in passing to hypercovers. The supplied text gives no evidence that the equivalence has been proved or disproved.

References

Primary source

Andrew W. Macpherson, “An alternative to hypercovers”, arXiv:2008.11912 (2020).

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