nLab right Bousfield delocalization

Idea

A model category structure $M_1$ is a right Bousfield localization of a model structure $M_2$ on the same underlying category if $M_1$ and $M_2$ have the same fibrations and the weak equivalences of $M_1$ contain those of $M_2$

The notion of right Bousfield delocalization reverses this relation: $M_1$ is a right Bousfield delocalization of $M_2$ if $M_2$ is a right Bousfield localization of $M_1$.

Of course, the nontrivial task here is to establish interesting existence criteria for right Bousfield delocalizations.

Existence theorem

Theorem (Corrigan-Salter)

If $M_1$ and $M_2$ are two cofibrantly generated model category structures on the same category with coinciding classes of fibrations, then there is a third cofibrantly generated model structure $M_3$ with the same fibrations and whose weak equivalences are the intersection of weak equivalences in $M_1$ and $M_2$. This model structure is a right Bousfield delocalization of both $M_1$ and $M_2$.

(Corrigan-Salter 15)

Related entries

• right Bousfield localization

• colocalization

References

• Bruce Corrigan-Salter, Right delocalization of model categories, arXiv:1504.04545.