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 {{nobreak|<math>\mathcal U\to X</math>,}} one can show that if the space <math>X</math> 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 <math>X</math> 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 <math>n</math>-fold intersections of the sets of the given open cover <math>\mathcal U</math>, to allow the pairwise intersections of the sets in <math>\mathcal U=\mathcal U_0</math> to be covered by an open cover <math>\mathcal U_1</math>, and to let the triple intersections of this cover to be covered by yet another open cover <math>\mathcal U_2</math>, 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 <math>X</math> be a scheme and consider the category of schemes étale over <math>X</math>. A '''hypercover''' is a semisimplicial object <math>U_\bullet</math> of this category such that <math>U_0 \to X</math> is an étale cover and such that <math>U_{n+1} \to \left(\left(\operatorname{\mathbf{cosk}}_n:= \operatorname{cosk}_n\circ\operatorname{tr}_n\right) U_\bullet\right)_{n+1}</math> is an étale cover for every <math>n\geq 0</math>.
Here, <math>U_{n+1} \to \left(\operatorname{\mathbf{cosk}}_n U_\bullet\right)_{n+1}</math> is the limit of the diagram which has one copy of <math>U_i</math> for each <math>i</math>-dimensional face of the standard <math>n+1</math>-simplex (for <math>0 \leq i \leq n</math>), one morphism for every inclusion of faces, and the augmentation map <math>U_0 \to X</math> at the end. The morphisms are given by the boundary maps of the semisimplicial object <math>U_\bullet</math>.
==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 <math>X</math>, the category <math>HR(X)</math> 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.
==References== {{reflist}}
* {{cite book|last1=Artin|first1=Michael|author-link1=Michael Artin|title=Etale homotopy|year=1969|publisher=Springer|last2=Mazur|first2=Barry|author-link2=Barry Mazur}} * {{cite book|last=Friedlander|first=Eric|author-link=Eric Friedlander|title=Étale homotopy of simplicial schemes|year=1982|publisher=Annals of Mathematics Studies, PUP}} * Lecture notes by G. Quick "[https://www.mathi.uni-heidelberg.de/fg-sga/docs/Etale%20Homotopy%20-%20Lecture%202.pdf Étale homotopy lecture 2]." * {{nlab|id=hypercover|title=Hypercover}}
{{Topology}}
Category:Homotopy theory