A Projective Model Structure on Pro Simplicial Sheaves, and the Relative \'Etale Homotopy Type

TL;DR

This paper introduces a new model structure on pro-simplicial sheaves to study étale homotopy, extending classical constructions to a relative setting and generalizing obstructions to rational points over schemes.

Contribution

It develops a novel model structure on pro-simplicial sheaves using weak fibration categories, enabling a natural extension of étale homotopy types to a relative context.

Findings

Constructed a new model structure on pro-simplicial sheaves.

Lifted Artin and Mazur's étale homotopy type in this framework.

Generalized homotopical obstructions from fields to schemes.

Abstract

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of \'etale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's \'etale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relative \'etale 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 pro-category, 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].