nLab right Bousfield localization of model categories

Idea

Right Bousfield localizations (alias cellularizations) of model categories are a way to present coreflective localizations of (∞,1)-categories via model categories.

Given a model category $M$, the formal definition and elementary properties can be deduced from the definition and properties of left Bousfield localizations of the opposite category of $M$.

However, the typical existence theorems for right Bousfield localizations are not formulated in this manner, since typical conditions involve some form of local presentability? of the underlying category, or an analogous condition, which typically do not hold for the opposite category.

Barwick’s existence theorem

If $C$ is a tractable model category, $D$ is an accessible accessibly embedded subcategory of $C$ that is closed under homotopy colimits in $C$. Then there is a right Bousfield localization of $C$, which is a right model category? (but not necessarily a model category), and its colocal objects are precisely the object of $D$.

Related concepts

• left Bousfield localization of model categories

References

The original article:

• A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150 [doi:10.1016/0040-9383(75)90023-3]

Detailed discussion (including existence results for left proper cellular model categories):

• Philip Hirschhorn, Chapter 5 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs 99 (2002) [ISBN:978-0-8218-4917-0, pdf toc, pdf]

Existence results for combinatorial model categories:

• Clark Barwick: On left and right model categories and left and right Bousfield localizations, Homology, Homotopy and Applications 12 2 (2010) 245–320 [doi:10.4310/hha.2010.v12.n2.a9, euclid:1296223884, subsuming: arXiv:0708.2067, arXiv:0708.2832, arXiv:0708.3435]

On right Bousfield localization specifically for stable model categories (such as model structures on spectra):

• David Barnes, Constanze Roitzheim: Stable left and right Bousfield localisations, Glasg. Math. J. 56 1 (2014) 13–42 [arXiv:10.1017/S0017089512000882, arXiv:1204.5384]

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