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

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

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

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.

7.4.1, to compute sheaf cohomology in arbitrary Grothendieck topologies.

For the étale site the definition is the following: Let

A hypercover is a semisimplicial object

is an étale cover and such that

is the limit of the diagram which has one copy of

), one morphism for every inclusion of faces, and the augmentation map

The morphisms are given by the boundary maps of the semisimplicial 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

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.