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Fibre sequences and localization of simplicial sheaves. (English) Zbl 1276.55020
A localization functor $$L_f$$ in a model category corresponds to a new model structure where the weak equivalences are the $$f$$-local equivalences. Nullification functors $$P_A$$ are those localizations for which the map $$f$$ one inverts is of the form $$A \rightarrow \star$$ and they determine a right proper model structure. A. J. Berrick and E. D. Farjoun studied the question of preservation of fiber sequences of spaces under localization, [Isr. J. Math. 135, 205-220 (2003; Zbl 1071.55008)]. They proved that a fiber sequence $$F \rightarrow E \rightarrow B$$ is preserved by $$P_A$$ if and only if the homotopy pull-back along the fiber of the nullification $$B \rightarrow P_A B$$ is trivial. A major ingredient here is fiberwise localization.
In the article under review the author provides a generalization of this criterion in the category of simplicial sheaves over a site, with applications to $$\mathbb A^1$$-homotopy theory when the site is that of smooth finite type schemes over a field. In this setting quasi-fibrations are replaced by Rezk’s sharp maps, or equivalently Jardine’s universally $$f$$-local maps, i.e., maps in the $$f$$-local model structure which behave well under pull-backs. A nice achievement is a construction of fiberwise localization for simplicial sheaves based on a homotopy colimit decomposition of the base by simplices $$\Delta^n \times U$$, with $$U$$ in the site. This is analogous to a topological construction, but different since $$U$$ is not contractible in general. Once this is done, the main result provides a characterization for universally $$f$$-local maps in the spirit of Berrick and Dror Farjoun. It holds for right proper model structures, in particular for nullifications, but possibly other localization functors.
Background and comparison of different viewpoints are given, which makes it a valuable reference on the subject.

##### MSC:
 55R65 Generalizations of fiber spaces and bundles in algebraic topology 55P60 Localization and completion in homotopy theory 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14F42 Motivic cohomology; motivic homotopy theory
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