(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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. (…)
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.
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.
Original articles include
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
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
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.
Discussion in positive characteristic is in
Étale homotopy type of moduli stacks of curves is discussed in