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arxiv: math/0106158 · v2 · submitted 2001-06-19 · 🧮 math.KT · math.AT

Etale realization on the A¹-homotopy theory of schemes

classification 🧮 math.KT math.AT
keywords homotopycategoryetaleschemesdefinitionfunctorpro-spacessimplicial
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We compare Friedlander's definition of the etale topological type for simplicial schemes to another definition involving realizations 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.

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