A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type
Abstract
In this work we shall introduce a new model structure on the category of prosimplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a prospace associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a proobject associated to a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [HaSc] for the study of obstructions to the existence of rational points. This relative notion enables to generalize these homotopical obstructions from fields to general base schemas 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 we call a "weak fibration category". Our theorem says that a weak fibration category can be "completed" into a full model category structure on its procategory, provided it satisfies some additional technical requirements. Our model structure is obtained by applying this result to the weak fibration category of simplicial sheaves over a Grothendieck site, where the weak equivalences and the fibrations are local in the sense of Jardine [Jar].
 Publication:

arXiv eprints
 Pub Date:
 September 2011
 arXiv:
 arXiv:1109.5477
 Bibcode:
 2011arXiv1109.5477B
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Algebraic Geometry
 EPrint:
 To appear in Advances in Mathematics