I am interested in the compact objects of various categories of categories. For example, $Cat^{small}$ is presentable and has compact objects that are retracts of finite colimits of $\Delta^n$, the $n$-simplex viewed as a category. To see this, note $\Delta^n$ are all compact and any small category is the canonical colimit of $\Delta^n$'s as $$ C \cong colim^{Cat}(Hom_{Cat}(\Delta^*, C)\rightarrow \Delta^* \rightarrow Cat^{small}) $$ Hence any category is a filtered colimit of a finite colimit of $\Delta^n$. So any compact object is a retract such a finite colimit.
In the case of $Cat^{small, stable}$, the category of small stable $\infty$-categories, Blumberg, Gepner, Tabuada https://arxiv.org/abs/1001.2282 showed that this category is presentable and compactly generated.
Question: what are compact objects of $Cat^{small, stable}$?
I suspect that compact objects are given by retracts of finite colimits of $Perf A_n = Fun(A_n, Sp)^c$. I think $Perf A_n$ is compact since the functor $Cat \rightarrow Cat^{small, stable}$ preserves compact objects since the adjoint preserves filtered colimits (what is a reference for this?) and hence finite colimits of $Perf A_n$ are also compact. Furthermore, I think that any small stable category $C$ is the canonical colimit of $Perf A_n$ as $C \cong colim^{Cat^{stable}}(Hom(Perf A_*, C) \rightarrow Perf A_* \rightarrow Cat^{small, stable})$. Again, I am not sure of a reference for this but I believe it is true since $Hom_{Cat^{stable}}(Perf A_n, C) \cong Hom_{Cat}(A_n, C)$.
I also think that there is a simpler description of the compact objects of $Cat^{stable, small}$.
Question: Is it true that any compact object $C$ in $Cat^{stable, small}$ is an essential localization of $Fun(C_0^{op}, Sp)^c$ for $C_0 \in Cat^{small}$ that is compact in $Cat^{small}$?
Note that $Fun(C_0^{op}, Sp)^c$ is compact in $Cat^{stable, small}$ since $C_0 \in Cat^{small}$ is compact and $Cat^{small} \rightarrow Cat^{stable, small}$ preserves colimits. So its retracts are also compact. The retracts of $Fun(C_0^{op}, Sp)^c$ arise from essential localizations, i.e localizations so that the fully faithful right adjoint has a right adjoint as well. On the other hand, by definition, any presentable stable category should be a reflective localization of $Fun(C^{op}, Sp)$ for a small category $C$. This suggests that any that small stable category should be a reflective localization of $Fun(C^{op}_0, Sp)^c$ for some small category $C_0$. Hence it seems reasonable that the compacts should be essential localizations of $Fun(C_0^{op}, Sp)^c$ for a compact $C_0$, as in the above question.
Finally:
Question: what are the compact objects of $PrL$, the category of presentable $(\infty,1)$-categories?
As discussed here Is the $\infty$-category of presentable $\infty$-categories presentable?, $PrL$ is not presentable. I suspect that $PrL$ doesn't have any compact objects. For example, it is not true that $Cat \rightarrow PrL$ preserves compact objects and the unit $Space \in PrL$ is not compact. On the other hand, if we view $PrL$ as a $PrL$-enriched category, then $Space$ is compact. So compactness depends on the enrichment category.
