Descent along atlases is equivalent to hyperdescent
Descent along atlases is equivalent to hyperdescent
Let be an -category with a Grothendieck topology, let be an object of , and let be an -presheaf. An atlas for an object is an -functor satisfying the covering condition specified in the paper, and satisfies descent along atlases if every atlas of every open induces an equivalence
Atlas descent conjecture. The following conditions on are equivalent: (1) satisfies descent along atlases; (2) satisfies hyperdescent. In particular, the full subcategory of 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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