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## Summary

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

## 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 $X$  be a scheme and consider the category of schemes étale over $X$ . A hypercover is a simplicial object $U_{\bullet }$  of this category such that $U_{0}\to X$  is an étale cover and such that $U_{n+1}\to (\operatorname {cosk} _{n}\operatorname {sk} _{n}U_{\bullet })_{n+1}$  is an étale cover for every $n\geq 0$ .

Here, $(\operatorname {cosk} _{n}\operatorname {sk} _{n}U_{\bullet })_{n+1}$  is the limit of the diagram which has one copy of $U_{i}$  for each $i$ -dimensional face of the standard $n+1$ -simplex (for $0\leq i\leq n$ ), and one morphism for every inclusion of faces. The morphisms are given by the boundary maps of the simplicial object $U_{\bullet }$ .

## 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 $X$ , the category $HR(X)$  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.