ABSTRACT
Morel–Voevodsky's A1-homotopy theory transports tools from algebraic topology into arithmetic and algebraic geometry, allowing us to draw arithmetic conclusions from topological arguments. Comparison results between classical and A1-homotopy theories can also be used in the reverse direction, allowing us to infer topological results from algebraic calculations. This chapter discusses realization functors to topological spaces, which allow us to see how A1-homotopy theory combines phenomena associated to the real and complex points of a variety. The standard choice of Grothendieck topology for A1-homotopy theory is the Nisnevich topology, although the etale topology is also used, producing a different homotopy theory of spaces. The etale realization functor can transport theorems in A1-homotopy theory to the older and very successful theory of étale homotopy or cohomology.
