In mathematics, especially in

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme ''X'', the idea is to consider étale coverings

U \rightarrow X

and to replace each connected component of ''U'' and the higher "intersections", i.e., fiber products,

U_n := U \times_X U \times_X \dots \times_X U

(''n''+1 copies of ''U'',

n \geq 0

) by a single point. This gives a simplicial set which captures some information related to ''X'' and the étale topology of it. Slightly more precisely, it is in general necessary to work with étale

hypercover 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 X is compact and if every in ...

(U_n)_

instead of the above simplicial scheme determined by a usual étale cover. Taking finer and finer hypercoverings (which is technically accomplished by working with the

pro-object In mathematics, the ind-completion or ind-construction is the process of freely adding filtered colimits to a given category ''C''. The objects in this ind-completed category, denoted Ind(''C''), are known as direct systems, they are functors from ...

in simplicial sets determined by taking all hypercoverings), the resulting object is the étale homotopy type of ''X''. Similarly to classical topology, it is able to recover much of the usual data related to the étale topology, in particular the

étale fundamental group The étale or algebraic fundamental group is an analogue in algebraic geometry, for schemes, of the usual fundamental group of topological spaces. Topological analogue/informal discussion In algebraic topology, the fundamental group ''π''1(''X' ...

of the scheme and the étale cohomology of locally constant étale sheaves.

References

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External links

*http://ncatlab.org/nlab/show/étale+homotopy {{DEFAULTSORT:Etale homotopy type Homotopy theory Algebraic geometry