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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.

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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.

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    $\begingroup$ 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. $\endgroup$
    – Simon Henry
    Apr 29, 2020 at 13:16
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    $\begingroup$ So, there is an example in any model of ZFC. Can one do it without mentioning VP at all? $\endgroup$
    – Jiří Rosický
    Apr 29, 2020 at 15:48

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