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Let $Pr^L$ denote the $\infty$-category of presentable $\infty$-categories and left adjoint functors. Let $Pr^L_\kappa \subset Pr^L$ denote the locally-full subcategory of $\kappa$-compactly-generated $\infty$-categories and left adjoints which preserve $\kappa$-compact objects (where $\kappa$ is some fixed regular cardinal). Note that in addition to being a subcategory of $Pr^L$, the $\infty$-category $Pr^L_\kappa$ is itself presentable, and so is naturally regarded as an object of $Pr^L$.

I would like to regard $Pr^L_\kappa \in Pr^L$ as an object which is analogous to an object classifier (a.k.a. "universe") $\mathcal U_\kappa$ in an $\infty$-topos $\mathcal E$. I'm not sure this is a "correct" intuition, so here are some questions which I hope will help sort out whether it is:

Question 1: What does the object $Pr^L_\kappa \in Pr^L$ represent?

That is, is there some snappy description of the hom-functor $Map_{Pr^L}(-, Pr^L_\kappa) : (Pr^L)^{op} \to SPACES$, or the 2-categorical hom-functor $Fun^L(-,Pr^L_\kappa) : (Pr^L)^{op} \to CAT$ or maybe the internal-hom functor $Fun^L(-, Pr^L_\kappa) :(Pr^L)^{op} \to Pr^L$? Ideally such a description would be along the lines of the functor represented by an object classifier, sending $\mathcal C \in Pr^L$ to some sort of size-restricted version of the slice $Pr^L/\mathcal C$ -- maybe with extra assumptions on $\kappa$ such as asking it to be strongly inaccessible.

Question 2: What does $Pr^L_\kappa \in Pr^L$ represent when restricted to $Pr^L_\kappa \subset Pr^L$?

That is, is there a nice description of the restricted hom-functor $Map_{Pr^L}(-,Pr^L_\kappa) : (Pr^L_\kappa)^{op} \to SPACES$, or of the 2-categorical or internal variants valued in $CAT$ or $Pr^L$?

Finally, observe that the $\infty$-category $Pr^L_\kappa$ (which is equivalent to the $\infty$-category $Rex_\kappa$ of small, idempotent-complete, $\kappa$-cocomplete categories and $\kappa$-cocontinuous functors) is itself $\kappa$-compactly-generated. So we may ask:

Question 3: What does $Pr^L_\kappa \in Pr^L_\kappa$ represent?

There are a few ways to make this precise. The functor $U$ "represented" by $Pr^L_\kappa$ might be taken to be any of the following:

  1. $U = Map_{Pr^L_\kappa}(-,Pr^L_\kappa) : (Pr^L_\kappa)^{op} \to Spaces$

  2. $U = Fun^{L,\kappa}(-,Pr^L_\kappa) : (Pr^L_\kappa)^{op} \to Cat$

  3. $U = Ind_\kappa(Fun^{L,\kappa}(-,Pr^L_\kappa)) : (Pr^L)_\kappa^{op} \to CAT$

  4. $U = Ind_\kappa(Fun^{L,\kappa}(-,Pr^L_\kappa)) : (Pr^L)_\kappa^{op} \to Pr^L_\kappa$

  5. $U = Ind_\kappa(Fun^{L,\kappa}(-,Pr^L_\kappa)) : (Pr^L)_\kappa^{op} \to Pr^L$

Of these, (4) looks particularly attractive to me, since it's the "internal" version. But perhaps another option (or something else entirely!) is actually a better way to think about it!

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    $\begingroup$ I think the case of colimit preserving functors $C \to Pr^L$ has a nice fibrational description. I would say something like a "left and right adjoint bifibration $D \to C$ that satisfies a descent condition with respect to colimit in $C$" (using that colimit in $Pr^L$ are limits in $Pr^R$). So, it might be a good idea to start by figuring out this case in details and then see what happen when one adds the further condition to take values in $Pr^L_\kappa$ and to preserves $\kappa$-compact objects... $\endgroup$
    – Simon Henry
    Commented Jul 15, 2022 at 16:29
  • $\begingroup$ @SimonHenry Ah, I think you're right... A colimit-preserving functor $F : C \to Pr^L$ such that the composite $C \to Pr^L \to CAT$ preserves $\kappa$-filtered colimits for some $\kappa$ is exactly a presentable fibration, and Gepner - Haugseng - Nikolaus show that in this case the Grothendieck construction $\int F$ of the functor is presentable, and the fibration $\int F \to C$ is an accessible functor. I don't think that $\int F \to C$ need be a left adjoint though, which is interesting -- we seem to get something bigger than the slice... $\endgroup$
    – Tim Campion
    Commented Jul 15, 2022 at 16:55
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    $\begingroup$ Under the condition you have it is both left and right adjoint: taking each object to the initial or terminal object of the fibers gives both adjoint. $\endgroup$
    – Simon Henry
    Commented Jul 15, 2022 at 16:58
  • $\begingroup$ Oh of course... So it appears that $Pr^L$, if it lived in $Pr^L$, would classify precisely cocartesian fibrations in $Pr^L$ in the usual 2-categorical sense. So $Pr^L_\kappa$ should definitely be some kind of "size-restricted fibration classifier". $\endgroup$
    – Tim Campion
    Commented Jul 15, 2022 at 17:05
  • $\begingroup$ Well, it's a bit more complicated than cocartesian fibration. I'm sure what the exact condition is, but you at least want bifibration with both adjoint, and you need a descent condition to express préservation of colimits. $\endgroup$
    – Simon Henry
    Commented Jul 15, 2022 at 17:51

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