nLab right Bousfield localization of model categories

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Idea

Right Bousfield localizations (alias cellularizations) of model categories are a way to encode coreflective localizations of (∞,1)-categories in the setting of model categories.

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

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

References

The original article:

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

Existence results for combinatorial model categories:

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

  • D. Barnes and C. Roitzheim. Stable left and right Bousfield localisations. Glasg. Math. J., 56(1):13–42, 2014.

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Created on December 28, 2023 at 22:11:22. See the history of this page for a list of all contributions to it.