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 U → X {\displaystyle {\mathcal {U}}\to X} , one can show that if the space X {\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 X {\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 n {\displaystyle n} -fold intersections of the sets of the given open cover U {\displaystyle {\mathcal {U}}} , to allow the pairwise intersections of the sets in U = U 0 {\displaystyle {\mathcal {U}}={\mathcal {U}}_{0}} to be covered by an open cover U 1 {\displaystyle {\mathcal {U}}_{1}} , and to let the triple intersections of this cover to be covered by yet another open cover U 2 {\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.
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 {\displaystyle X} be a scheme and consider the category of schemes étale over X {\displaystyle X} . A hypercover is a semisimplicial object U ∙ {\displaystyle U_{\bullet }} of this category such that U 0 → X {\displaystyle U_{0}\to X} is an étale cover and such that U n + 1 → ( ( c o s k n := cosk n ∘ tr n ) U ∙ ) n + 1 {\displaystyle U_{n+1}\to \left(\left(\operatorname {\mathbf {cosk} } _{n}:=\operatorname {cosk} _{n}\circ \operatorname {tr} _{n}\right)U_{\bullet }\right)_{n+1}} is an étale cover for every n ≥ 0 {\displaystyle n\geq 0} . Here, U n + 1 → ( c o s k n U ∙ ) n + 1 {\displaystyle U_{n+1}\to \left(\operatorname {\mathbf {cosk} } _{n}U_{\bullet }\right)_{n+1}} is the limit of the diagram which has one copy of U i {\displaystyle U_{i}} for each i {\displaystyle i} -dimensional face of the standard n + 1 {\displaystyle n+1} -simplex (for 0 ≤ i ≤ n {\displaystyle 0\leq i\leq n} ), one morphism for every inclusion of faces, and the augmentation map U 0 → X {\displaystyle U_{0}\to X} at the end. The morphisms are given by the boundary maps of the semisimplicial object U ∙ {\displaystyle U_{\bullet }} .
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