The Čech descent condition is used to refer to a specific ∞-descent condition for ∞-presheaves.
Concretely, Čech descent can be implemented by performing the left Bousfield localization of the model category of simplicial presheaves equipped with the projective model structure (say) at Čech nerves of covering families. Then fibrant objects in the resulting model structure will be precisely presheaves of Kan complexes that satisfy the Čech descent condition.
Čech descent should be contrasted with hyperdescent. The hyperdescent condition is stronger than the Čech descent condition, since it uses arbitrary hypercovers instead of Čech nerves of covering families. 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 20:26:26. See the history of this page for a list of all contributions to it.