In mathematics, especially in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the étale homotopy type is an analogue of the homotopy type of

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s for

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

. 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 In mathematics, a simplicial set is a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...

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 hypercovers

(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 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 of the scheme and the

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

of locally constant étale sheaves.

Example

For

Spec(k)

, morally taking

Spec(\bar)

the simplicial complex would be the one computing

BGal(k)

Modern perspective

The pro-etale site gives a refinement and generates a homotopical object in simplicial profinite sets. This has a

forgetful functor In mathematics, more specifically in the area of category theory, a forgetful functor (also known as a stripping functor) "forgets" or drops some or all of the input's structure or properties mapping to the output. For an algebraic structure of ...

to the

homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed ...

of profinite simplicial sets.

References

* * * Scholze, pro-etale site. https://www.youtube.com/watch?v=qL5uxXAeU00&ab_channel=HausdorffCenterforMathematics

External links

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