nLab
étale homotopy

Context

Étale morphisms

Homotopy theory

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

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

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 XX a paracompact topological space and {U iX}\{U_i \to X\} a good cover of XX 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 XX.

If XX 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 XX.

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

Example

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

(HSS 13, section 1)

Proposition

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

(HSS 13, theorem 1)

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 June 7, 2014 04:30:47 by Urs Schreiber (89.204.130.218)