nLab étale homotopy

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.

(HSS 13, section 1)

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.

(HSS 13, theorem 1)

Related concepts

• Čech homology

• algebraic fundamental group

• anabelian geometry

• étale topos, pro-étale topos

• étale cohomology

References

General

Original articles include

• Michael Artin, Barry Mazur, Homotopy of varieties in the etale topology, Proceedings of a Conference on Local Fields, Driebergen 1966, Springer.

• Michael Artin, Barry Mazur, Étale Homotopy, Springer Lecture Notes in Mathematics 100, Berlin (1969)

• Eric Friedlander, Fibrations in étale homotopy theory (numdam), Publ. Math.

  Inst. des Haut. Études Scient., 42, (1973), 5 – 46.

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

• Daniel Isaksen, Étale realization of the $\mathbb{A}^1$-homotopy theory of schemes, 2001 (K-theory archive)

• G. Quick, Stable étale realization and étale cobordism, Adv. in Math.,

  214 (2007), 730–760.

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

• Marc Hoyois, Higher Galois theory, (pdf)

and (Hoyois 13b, section 1).

and in, where a comparison theorem over the complex numbers is also proven:

• Dave Carchedi, On the étale homotopy type of higher stacks (arXiv:1511.07830)

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

• James Milne, section 4 of Lectures on Étale Cohomology

  Generalization to simplicial schemes is discussed in

• Eric Friedlander, 1982, Étale homotopy of simplicial schemes , volume 104 of Annals of Mathematics Studies , Princeton University Press, Princeton, N.J.

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

• Marc Hoyois, The étale-symmetric Künneth theorem, 2013 (pdf)

Discussion in positive characteristic is in

• Armin Holschbach, Johannes Schmidt, Jakob Stix, Étale contractible varieties in positive characteristic (arXiv:1310.2784)

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)