A projective model structure on pro-simplicial sheaves, and the relative étale homotopy type

Ilan Barnea, Tomer M. Schlank*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 étale homotopy. Using this model structure we define a pro-space associated with a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [3] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated with a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [18] 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 [23].

Original languageAmerican English
Pages (from-to)784-858
Number of pages75
JournalAdvances in Mathematics
StatePublished - 19 Mar 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 Elsevier Inc.


  • Model category
  • Pro-category
  • Simplicial presheaves
  • Étale homotopy


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