Abstract
Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.
Highlights
Localization techniques play an important role in modern homotopy theory
Given a diagram of model categories F : Iop → CAT, there is an injective model structure on the category of sections associated wit F, which we can further colocalize to obtain the homotopy limit model structure. We study these model structures for towers and homotopy fibered products of model categories
We show that the category of symmetric spectra is Quillen equivalent to the homotopy limit model structure of the left Quillen presheaf for Bousfield arithmetic squares of spectra
Summary
Localization techniques play an important role in modern homotopy theory. For several applications it is often useful to approximate a given space or spectrum by simpler ones by means of localization functors. Given a diagram (left Quillen presheaf) of model categories F : Iop → CAT, there is an injective model structure on the category of sections associated wit F , which we can further colocalize to obtain the homotopy limit model structure. We study these model structures for towers and homotopy fibered products (homotopy pullbacks) of model categories. As an application we show that for simplicial sets and for bounded below chain complexes these towers converge in a certain sense Another tower model structure is the homotopy limit model structure on the left Quillen presheaf of chromatic towers Chrom(Sp), where Sp denotes here the category of p-local symmetric spectra. Vol 13 (2016) Towers and Fibered Products of Model Structures 3865 layers of the Postnikov towers established earlier and to study the correspondence between stable localizations and stable colocalizations
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