In this work we shall introduce a new model structure on the category ofpro-simplicial sheaves, which is very convenient for the study of \'etalehomotopy. Using this model structure we define a pro-space associated to atopos, as a result of applying a derived functor. We show that our constructionlifts Artin and Mazur's \'etale homotopy type [AM] in the relevant specialcase. Our definition extends naturally to a relative notion, namely, apro-object associated to a map of topoi. This relative notion lifts therelative \'etale homotopy type that was used in [HaSc] for the study ofobstructions to the existence of rational points. This relative notion enablesto generalize these homotopical obstructions from fields to general baseschemas and general maps of topoi. Our model structure is constructed using a general theorem that we prove.Namely, we introduce a much weaker structure than a model category, which wecall a "weak fibration category". Our theorem says that a weak fibrationcategory can be "completed" into a full model category structure on itspro-category, provided it satisfies some additional technical requirements. Ourmodel structure is obtained by applying this result to the weak fibrationcategory of simplicial sheaves over a Grothendieck site, where the weakequivalences and the fibrations are local in the sense of Jardine [Jar].
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