2011/09/26 by Ilan Barnea, Barnea, Ilan, Tomer M. Schlank +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1109.5477
openalex publication_date 2011/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we shall introduce a new model structure on the category of\npro-simplicial sheaves, which is very convenient for the study of 'etale\nhomotopy. Using this model structure we define a pro-space associated to a\ntopos, as a result of applying a derived functor. We show that our construction\nlifts Artin and Mazur's 'etale homotopy type [AM] in the relevant special\ncase. Our definition extends naturally to a relative notion, namely, a\npro-object associated to a map of topoi. This relative notion lifts the\nrelative 'etale homotopy type that was used in [HaSc] for the study of\nobstructions to the existence of rational points. This relative notion enables\nto generalize these homotopical obstructions from fields to general base\nschemas and general maps of topoi.\n Our model structure is constructed using a general theorem that we prove.\nNamely, we introduce a much weaker structure than a model category, which we\ncall a "weak fibration category". Our theorem says that a weak fibration\ncategory can be "completed" into a full model category structure on its\npro-category, provided it satisfies some additional technical requirements. Our\nmodel structure is obtained by applying this result to the weak fibration\ncategory of simplicial sheaves over a Grothendieck site, where the weak\nequivalences and the fibrations are local in the sense of Jardine [Jar].\n