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, CartSp, Stein site, and most of the other sites used in differential and complex geometry, the hyperdescent condition coincides with the Čech descent condition.
Created on February 4, 2023 at 20:30:13. See the history of this page for a list of all contributions to it.