Towers and fibered products of model structures

TL;DR

This paper investigates the homotopy limit model structures arising from towers and fibered products of model categories, with applications to Postnikov towers, chromatic towers, and Bousfield squares, revealing new stability properties.

Contribution

It introduces a framework for analyzing homotopy limit model structures in towers and fibered products, with novel results on stability in Bousfield localizations.

Findings

Homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization

Applications to Postnikov and chromatic towers of spectra

Analysis of Bousfield arithmetic squares of spectra

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.