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 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.