In mathematics, and in particular homotopy theory, a hypercovering (or hypercover) is a simplicial object that generalises the Čech nerve of a cover. For the Čech nerve of an open cover , one can show that if the space is compact and if every intersection of open sets in the cover is contractible, then one can contract these sets and get a simplicial set that is weakly equivalent to in a natural way. For the étale topology and other sites, these conditions fail. The idea of a hypercover is to instead of only working with -fold intersections of the sets of the given open cover , to allow the pairwise intersections of the sets in to be covered by an open cover , and to let the triple intersections of this cover to be covered by yet another open cover , and so on, iteratively. Hypercoverings have a central role in étale homotopy and other areas where homotopy theory is applied to algebraic geometry, such as motivic homotopy theory.
The original definition given for étale cohomology by Jean-Louis Verdier in SGA4, Expose V, Sec. 7, Thm. 7.4.1, to compute sheaf cohomology in arbitrary Grothendieck topologies. For the étale site the definition is the following:
Here, is the limit of the diagram which has one copy of for each -dimensional face of the standard -simplex (for ), and one morphism for every inclusion of faces. The morphisms are given by the boundary maps of the simplicial object .
The Verdier hypercovering theorem states that the abelian sheaf cohomology of an étale sheaf can be computed as a colimit of the cochain cohomologies over all hypercovers.
For a locally Noetherian scheme , the category of hypercoverings modulo simplicial homotopy is cofiltering, and thus gives a pro-object in the homotopy category of simplicial sets. The geometrical realisation of this is the Artin-Mazur homotopy type. A generalisation of E. Friedlander using bisimplicial hypercoverings of simplicial schemes is called the étale topological type.