Hypercovering

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 formula_1 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 formula_1 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 formula_3-fold intersections of the sets of the given open cover formula_4, to allow the pairwise intersections of the sets in formula_5 to be covered by an open cover formula_6, and to let the triple intersections of this cover to be covered by yet another open cover formula_7, 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.
Formal definition.
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:
Let formula_1 be a scheme and consider the category of schemes étale over formula_1. A hypercover is a simplicial object formula_10 of this category such that formula_11 is an étale cover and such that formula_12 is an étale cover for every formula_13.
Here, formula_14 is the limit of the diagram which has one copy of formula_15 for each formula_16-dimensional face of the standard formula_17-simplex (for formula_18), and one morphism for every inclusion of faces. The morphisms are given by the boundary maps of the simplicial object formula_10.
Properties.
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 formula_1, the category formula_21 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.