# nLab étale homotopy

## Theorems

#### $(\infty,1)$-Topos Theory

(∞,1)-topos theory

## Constructions

structures in a cohesive (∞,1)-topos

# Contents

## Idea

The notion of étale homotopy can be understood as a vast generalization of the following classical fact.

The nerve theorem says that for $X$ a paracompact topological space and $\{U_i \to X\}$ a good cover of $X$ by open subsets, then the simplicial set obtained from the Cech nerve of the covering by degreewise contracting all connected components to a point, presents the homotopy type of $X$.

If $X$ here is more generally a locally contractible space there is in general no notion of “good” enough open cover anymore. Instead, one can consider the above kind of construction for all hypercovers and take the limit over the resulting simplicial sets. The classical theorem by Artin-Mazur states that this still gives the homotopy type of $X$.

The construction itself, however, makes sense for arbitrary topological spaces and in fact for arbitrary sites.

In the literature, particularly the étale site is often considered and “étale homotopy” is often implicitly understood to take place over this site.

But the concept is much more general. In particular, one can understand the construction of the limit over contractions of hypercovers as a presentation of naturally defined (∞,1)-functors in (∞,1)-topos theory.

Notably, if the given site is a a locally ∞-connected site, then the étale homotopy construction computes precisely the derived functor that presents the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos. Many constructions in the literature can be understood as being explicit realizations of this simple general concept. Detailed discussion of this is at geometric homotopy groups in an (∞,1)-topos.

Even more generally, étale homotopy give the notion of shape of an (∞,1)-topos. (…)

## Examples

### Chevalley groups and Galois groups

For the special case of fundamental groups, the concept of étale homotopy groups also goes by the name of Chevalley fundamental groups.

The étale fundamental group of a scheme is its absolute Galois group. See at Galois theory – Statement of the main result.

### Étale contractibility

###### Example

For $k$ a field of characteristic 0, then the affine line $\mathbb{A}^1_k$ is étale contractible. This is no longer the case in positive characteristic.

###### Proposition

Let $k$ be an algebraically closed field of positive characteristic. Then the only smooth variety over $k$ which is étale contractible is the point $Spec(k)$. In fact this is the only smooth variety which is 2-connected.

## References

### General

Original articles include

The modern perspective from the point of view of model structures on simplicial presheaves is in

and fully abstractly from the point of view of (∞,1)-topos-theory (shape of an (∞,1)-topos) in

and (Hoyois 13b, section 1).

An introduction is in

• Tomer Schlank, Alexei Skorobogatov, A very brief introduction to étale homotopy. In: “Torsors, étale homotopy and applications to rational points”. LMS Lecture Note Series 405, Cambridge University Press, 2013. (pdf)

Lecture notes on the étale fundamental group are in

More on this is in

• Michael Misamore, Étale homotopy types and bisimplicial hypercovers, Homology, Homotopy and Applications, Vol. 15 (2013), No. 1, pp.27-49. (web)

### Examples and applications

Discussion in positive characteristic is in

Étale homotopy type of moduli stacks of curves is discussed in

• Paola Frediani, Frank Neumann, Étale homotopy types of moduli stacks of algebraic curves with symmetries, K-Theory 30: 315-340, 2003 (arXiv:math/0404387)

Revised on September 1, 2014 07:37:55 by Urs Schreiber (82.113.98.44)