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 ${\displaystyle {\mathcal {U}}\to X}$, one can show that if the space ${\displaystyle X}$ 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 ${\displaystyle X}$ 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 ${\displaystyle n}$-fold intersections of the sets of the given open cover ${\displaystyle {\mathcal {U}}}$, to allow the pairwise intersections of the sets in ${\displaystyle {\mathcal {U}}={\mathcal {U}}_{0}}$ to be covered by an open cover ${\displaystyle {\mathcal {U}}_{1}}$, and to let the triple intersections of this cover to be covered by yet another open cover ${\displaystyle {\mathcal {U}}_{2}}$, 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.