Localization theories for simplicial presheaves. (English) Zbl 0914.55004

The authors introduce new theories: localizations for simplicial presheaves and presheaves of spectra at homology theories represented by presheaves of spectra and a theory of localization along a geometric topos morphism. The closed model structure for simplicial presheaves, a general \(f\)-localization theory for simplicial presheaves, the closed model structures of various stable categories and a homology localization technique for presheaves of spectra, all arise from a simple collection of axioms for classes of cofibrations and weak equivalences (axioms E1–E7 and sE1–sE7 for spectra). They define a localization theory as a closed model structure on the category of simplicial presheaves on a site \(C\) which arises from a class \(E\) of cofibrations which satisfy the conditions E1–E7. The authors specialize this theory to the standard examples of localization theory in ordinary homotopy theory. The standard homotopy theory of simplicial presheaves and sheaves on an arbitrary small Grothendieck site \(C\) is a type of localization theory. New theories have been discovered: a notion of localization along a geometric topos morphism and a resulting method of localization a space or a spectrum at a generalized homology theory arising from a presheaf of spectra on an arbitrary site. Application for these techniques is that of the \(f\)-localization results (with references to the work of Dror-Farjoun, Hirschhorn, Bousfield) presented in the fourth section of the paper.


55P60 Localization and completion in homotopy theory
19E08 \(K\)-theory of schemes
18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)
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