**Question:** What is an example of a locally presentable category $\mathcal C$ such that there exists a proper class of accessible localizations $(\mathcal C \to \mathcal D\_i)\_{i < ORD}$?


In other words, $(D\_i)\_{i < ORD}$ should be a proper class of full reflective subcategories of $\mathcal C$ which are accessibly embedded in $\mathcal C$.


I'm also interested in the infinity-categorical setting, though I suspect it doesn't make much difference.



Can a locally presentable category have a proper class of accessible localizations?
A limit closure of a set of objects of a locally presentable category $\mathcal K$ is reflective. In this way one gets an increasing chain of reflective full subcategories of $\mathcal K$. If this chain stops $\mathcal K$ has a cogenerator. Since a category of groups does not have a cogenerator, it has a proper class of reflective full subcategories. These subcategories are accessibly embedded under Vopěnka's principle. I do not expect that Vopěnka's principle is needed but, at this moment, I do not see how to avoid it.


