Etale realization on the A^1-homotopy theory of schemes
 
Daniel C. Isaksen
 
14F42 (primary), 14F35 (secondary)
 
Department of Mathematics
University of Notre Dame
Notre Dame, IN 46556
 
isaksen.1@nd.edu
 
We compare Friedlander's definition of etale homotopy for
simplicial schemes to another definition involving homotopy colimits of
pro-simplicial sets.  This can be expressed as a notion of hypercover
descent for etale homotopy.  We use this result to construct a homotopy
invariant functor from the category of simplicial presheaves on the etale
site of schemes over S to the category of pro-spaces. After completing
away from the characteristics of the residue fields of S, we get a functor
from the Morel-Voevodsky A^1-homotopy category of schemes to the homotopy
category of pro-spaces.