# Can a locally presentable category have a proper class of accessible localizations?

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.

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.
• Don't we also get easy example assuming the negation of Vopenka ? if A is a non-accessible localization of C, then we can consider the class of localization of C at any set S of morphisms inverted by the reflection $C \to A$. If these only forms a set, as $A$ identifies with the localization at their unions, it would make $A$ a localization at a set of arrows hence an accessible localization. – Simon Henry Apr 29 '20 at 13:16