Descent along atlases is equivalent to hyperdescent

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 VXV\subseteq X is an \infty-functor U:IT/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 VXV\subseteq X induces an equivalence

F(V)limi: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.

Sources & referencesView supporting material

Primary source

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

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