Presentable categories as colimits of finitely presentable categories

Question

I am trying to understand the relationship betweeen compactly generated presentable categories, also called finitely presentable categories, and general presentable categories (which I have less intuition for). A presentable $\infty$-category $C$ is an accessible reflective localization of a presheaf category on a small category $C_0$, i.e. $\pi: Fun(C_0^{op}, Space) \overset{\rightarrow}{\hookleftarrow} C: i$ where $i$ preserves $k$-filtered colimits for some $k$. The presheaf category is compactly generated and I think $C$ is compactly generated if and only if $i$ preserves filtered colimits. I think to meaningfully discuss reflective localizations in terms of morphisms in $PrL$, we need to consider $PrL$ as a $(\infty,2)$-category.

I am wondering if there are other ways to get to all presentable categories from compactly generated ones (that are more $(\infty,1)$-categorical). For example:

Question: Is every presentable category a (filtered?) colimit of compactly generated categories in $PrL$?

For example, if one could convert the reflective localization into a colimit in $PrL$, then there would be hope for this question to have an affirmative answer. My question may also be related to Why are compactly generated $\infty$-categories closed under limits in $\operatorname{Cat}_{\infty}$?, but rather for filtered colimits rather than filtered limits.

Answer 1

It is known (see e.g. Lemma 7.14 here) that $\mathrm{Pr}^\mathrm{L}$ is generated under colimits by $\mathcal{S}^{\Delta^1}$, which is certainly compactly generated (though as Victor points out, the functors that you use cannot all preserve compact objects, otherwise you stay in $\mathrm{Pr}^\mathrm{L}_\omega$).

I do not know, however, whether every $C\in \mathrm{Pr}^\mathrm{L}$ is canonically the colimit of compactly generated categories. It is a single colimit of compactly generated ones (as I'll explain below), but it is not clear to me whether $\mathrm{colim}_{D\in \mathrm{Pr}^\mathrm{L}_{(\omega)} / C} D$ exists, and if it does, whether it is equivalent to $C$ (here, I use $\mathrm{Pr}^\mathrm{L}_{(\omega)}$ to denote the full subcategory of $\mathrm{Pr}^\mathrm{L}$ spanned by compactly generated categories), or even whether this is an interesting question!

I also don't know the answer to the parenthetical question: can one make it a filtered colimit of compactly generated categories ?

To realize each $C$ as a single colimit of compactly generated categories, notice that "presentations" of $C$ in terms of localizations of presheaf categories equivalently give you presentations as pushouts: if $C$ is a localization of $\mathrm{Psh}(C_0)$ at the set of maps $S$, then it is equivalently the pushout $\mathrm{Psh}(S)\coprod_{\mathrm{Psh}(\coprod_S \Delta^1)} \mathrm{Psh}(C_0)$ - note, here, that $\Delta^1\to \mathrm{pt}$ is a localization, and hence applying $\mathrm{Psh}$ to it also is, and is therefore an epimorphism in $\mathrm{Pr}^{\mathrm{L}}$. This observation can allow you to prove this fact. Note that in this pushout, one of the legs preserves compacts.

At least stably, epimorphisms completely detect localizations in $\mathrm{Pr}^\mathrm{L}_{\mathrm{st}}$ and so there is no "real" need for the $2$-categorical structure to detect that (though I do not know if this is true unstably) - I discussed this in Appendix B here, the argument given in B.4 easily dualizes by observing how one computes pushouts in $\mathrm{Pr}^\mathrm{L}$)

Answer 2

It seems to me that the question you have linked answers this in the negative. Recall $\mathrm{Pr}^L\simeq(\mathrm{Pr}^R)^{\mathrm{op}}$, so that colimits in the former are computed by taking the adjoints of every functor in the diagram and then forming the limit (in either $\mathrm{Pr}^R$ or $\mathrm{Cat}_{\infty}$, they are the same). In particular, since by Maxime's answer, compactly generated presentable categories are closed under limits in $\mathrm{Pr}^R$, or equivalently $\mathrm{Cat}_{\infty}$, then they also must be closed under colimits in $\mathrm{Pr}^L$.

Edit: as Maxime points out in the comments, his answer only pertains to compact-preserving functors (aka filtered-colimit preserving right adjoints), so compactly-generated categories are only closed under compact-preserving functors.