A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type

Details A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type

2011-09-26

arxiv.org/abs/1109.5477v5

Mathematics

Algebraic Topology

We corrected a technical error, an added new stuff

Scientific paper

In his paper, Isaksen showed that a proper model category $\cC$, induces a model structure on the pro category $\Pro(\cC)$. In this paper we generalize Isaksen's theorem to the case when $\cC$ possess a weaker structure, which we call a "weak fibration category". Namely, we show that if $\cC$ is a weak fibration category, that satisfies an extra condition, there is a naturally induced model structure on $\Pro(\cC)$. We then apply our theorem to the case when $\cC$ is the weak fibration category of simplicial sheafs on a Grothendieck site, where both weak equivalences and fibrations are local as in Jardine. This gives a new model structure on the category of pro simplicial sheaves. 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, 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 by Harpaz and Schlank for the study of obstructions to the existence of rational points. Thus we embed the results of Harpaz and Schlank in a suitable model structure. This relative notion also enables to generalize these homotopical obstructions from fields to general base schemas and general maps of topoi.

Affiliated with

Also associated with

No associations

Rating

A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-688496