an:06261078
Zbl 1297.55020
Barnes, David; Roitzheim, Constanze
Stable left and right Bousfield localisations
EN
Glasg. Math. J. 56, No. 1, 13-42 (2014).
00328254
2014
j
55U35 55P42 55P60 18E30 16D90
Bousfield localization; stable model category
Many interesting model categories arise by enlarging the class of weak equivalences in a given (simpler) model category \(\mathcal C\). Often this is achieved by forming the \textit{left Bousfield localization} of \(\mathcal C\) with respect to a class of maps \(S\). Left Bousfield localizations are known to exist if \(\mathcal C\) and \(S\) satisfy suitable assumptions, see e.g. [\textit{P. S. Hirschhorn}, Model categories and their localizations. Mathematical Surveys and Monographs. 99. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1017.55001)].
In the paper under review, the authors define a class of maps \(S\) in a stable model category \(\mathcal C\) to be \textit{stable} if the \(S\)-local objects are closed under suspension. The main results state that if in this situation the left Bousfield localization exists, then it has various desirable properties: It is again a stable model structure, it is right proper if \(\mathcal C\) is, it has an explicit set of generating acyclic cofibrations if \(\mathcal C\) is proper and cellular, and it interacts well with monoidal products. The authors also establish corresponding results for right Bousfield localizations and discuss applications to Morita theory and spectral model categories.
Steffen Sagave (Bonn)
Zbl 1017.55001