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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 4th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## Idea The **hyperdescent** condition is used to refer to a specific [[∞-descent]] condition for [[∞-presheaves]]. Concretely, hyperdescent can be implemented by performing the [[left Bousfield localization]] of the [[model category]] of [[simplicial presheaves]] equipped with the [[projective model structure]] (say) at [[hypercovers]]. Equivalently, we can localize at those morphisms of [[simplicial presheaves]] that induce isomorphisms on all sheaves of homotopy groups. Then [[fibrant objects]] in the resulting [[model structure]] will be precisely [[presheaves]] of [[Kan complexes]] that satisfy the hyperdescent condition. Model structures on [[simplicial sheaves]] implementing hyperdescent were first constructed by [[André Joyal]] in mid-1980s. [[John F. Jardine]] then constructed analogous model structures for [[simplicial presheaves]]. Hyperdescent should be contrasted with [[Čech descent]]. The [[Čech descent]] condition is weaker than the hyperdescent condition, since it uses [[Čech nerves]] of [[covering families]] instead of arbitrary [[hypercovers]]. However, for [[hypercomplete sites]] such as [[Zariski site]], [[Nisnevich site]], [[Cartesian site]], [[Stein site]], and most of the other sites used in differential and complex geometry, the [[hyperdescent]] condition coincides with the Čech descent condition. ## Related concepts * [[Čech descent]] * [[∞-presheaf]], [[∞-sheaf]], [[simplicial presheaf]] * [[hypercover]] <a href="https://ncatlab.org/nlab/revision/hyperdescent/1">v1</a>, <a href="https://ncatlab.org/nlab/show/hyperdescent">current</a>

   Created:

   Idea

   The hyperdescent condition is used to refer to a specific ∞-descent condition for ∞-presheaves.

   Concretely, hyperdescent can be implemented by performing the left Bousfield localization of the model category of simplicial presheaves equipped with the projective model structure (say) at hypercovers. Equivalently, we can localize at those morphisms of simplicial presheaves that induce isomorphisms on all sheaves of homotopy groups. Then fibrant objects in the resulting model structure will be precisely presheaves of Kan complexes that satisfy the hyperdescent condition.

   Model structures on simplicial sheaves implementing hyperdescent were first constructed by André Joyal in mid-1980s. John F. Jardine then constructed analogous model structures for simplicial presheaves.

   Hyperdescent should be contrasted with Čech descent. The Čech descent condition is weaker than the hyperdescent condition, since it uses Čech nerves of covering families instead of arbitrary hypercovers. However, for hypercomplete sites such as Zariski site, Nisnevich site, Cartesian site, Stein site, and most of the other sites used in differential and complex geometry, the hyperdescent condition coincides with the Čech descent condition.

   Related concepts

   • Čech descent

   • ∞-presheaf, ∞-sheaf, simplicial presheaf

   • hypercover

   v1, current