On the étale homotopy type of Voevodsky's spaces , by Alexander Schmidt

In this paper we construct a natural extension of the functor `etale homotopy type' (defined by Artin and Mazur) from the category of smooth schemes over a field k to the category of simplicial étale sheaves on Sm/k. This functor factors through simplicial weak equivalences, thus induces a functor `etale homotopy type' on the simplicial homotopy category. Furthermore, if k has characteristic zero and finite virtual cohomological dimension, then the functor factors over weak A^1-equivalence. As a result, we can attach étale homotopy groups to any (pointed) object of the A^1-homotopy category of smooth schemes over k.

Alexander Schmidt <schmidt@mathi.uni-heidelberg.de >