What are the reflective subcategories of the category of presentable categories?

Question

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well.

A nice property of presentable $\infty$-categories is that if you have one such $\mathcal{C}$ and you have a full subcategory $\mathcal{D}\subseteq\mathcal{C}$ of it, then:

Theorem: The inclusion $\mathcal{D}\subseteq\mathcal{C}$ is reflective (i.e. has a left adjoint) if and only if $\mathcal{D}$ is closed under (small) limits and sufficiently filtered colimits.

This almost looks like the adjoint functor theorem, but note that we don't know that $\mathcal{D}$ is presentable in advance. In $1$-categories this is proved here, and I bet it is true $\infty$-categorically as well. Let's assume this for a moment.

Now, let $\mathbf{Pr}^L$ be the $\infty$-category of presentable $\infty$-categories and left adjoint (equivalently, colimit preserving) functors. I have a full subcategory $\mathcal{C}\subseteq \mathbf{Pr}^L$ and I want to know that there is a left adjoint to the inclusion, under some general closure conditions on $\mathcal{C}$. This would have been the case if only $\mathbf{Pr}^L$ was itself presentable...

Of course, there are some serious size issues here! The $\infty$-category $\mathbf{Pr}^L$ is huge (i.e. more than large, which is itself more than small) and I don't think it even has all large limits. What would be the poset $0 \to 1$ to the power of "a lot"? (at least its limit in the $\infty$-category of large categories is not presentable). Also, presentable $\infty$-categories are locally small, this doesn't seem to allow large limits. To conclude, I don't think $\mathbf{Pr}^L$ is "presentable" in the sense that we replace "small" with "large" and "large" with "huge". Keeping "small" and replacing only "large" with "huge" also doesn't seem to work for trivial reasons, since $\mathbf{Pr}^L$ can't be generated from a small set by small colimits, otherwise, it would have been at most large. Nevertheless, I still hope there might be a positive answer to the question in the title:

Question: Is there a characterization of reflective subcategories of $\mathbf{Pr}^L$ in terms of closure properties?

Perhaps this can be achieved by showing that $\mathbf{Pr}^L$ is "presentable" in some new sense, which takes into account the three sizes small-large-huge.

Answer 1

Some ideas, building off of Simon Henry and Ivan di Liberti's remarks:

1. $Pr^L$ is in fact essentially a large (not huge) category (in either the ordinary or $\infty$ context).

That is, let $\lambda$ be the size of (a skeleton of) the universe of small sets (i.e. "small" means $\lambda$-small, and $\lambda$ is inaccessible). Choose one representative from each equivalence class of objects of $Pr^L$ [1]. I claim that there are $\lambda$-many of these, that the number of isomorphism classes of morphisms between any two of them is $\lambda$, and between any two of these the space of natural isomorphisms (even natural transformations) has at most $\lambda$-many cells. All told, $Pr^L$ localized at the equivalences of categories is equivalent to a large quasicategory with $\lambda$-many simplices.

I've detailed a size estimate at the end.

2. $Pr^L$ has small limits and colimits.

They are easy to compute: in both $Pr^L$ and $Pr^R$, small limits are computed at the level of underlying categories.

3. $Pr^L$ is built from the presentable categories $Pr^L_\kappa \simeq \kappa-Cocts$ in the following manner.[3]

We have a large chain of presentable categories $Pr^L_\kappa$, and $Pr^L$ is the colimit. The linking functors $Pr^L_\kappa \to Pr^L_\mu$ are left adjoints, so preserve colimits, since the corresponding functor $\kappa-Cocts \to \mu-Cocts$ is given by $Ind_\kappa^\mu$, the free completion under $\mu$-small, $\kappa$-filtered colimits, which is left adjoint to the forgetful functor. Here $\kappa-Cocts$ is the category of small categories with $\kappa$-small colimits and functors that preserve them.

4. Generation properties.

One kind-of surprising consequence of this is that $Pr^L_\kappa$ is closed under small colimits in $Pr^L$ (not being a full subcategory, it is certianly not coreflective). In particular, $Pr^L$ is not generated under small colimits by any small subcategory. For example, even though every small category is a colimit of copies of the walking arrow category $[2]$, it's not the case that every object of $Pr^L$ is a colimit of copies of $Set^{[2]}$. (oops -- This doesn't rule out that it might be generated by a small set of objects, viewed as a (large) full subcategory!) On the other hand, each object $C$ of $Pr^L$ is a localization of a presheaf category $Psh(C_0) \in Pr^L_0$, and is in fact the geometric realization of a simplicial object $C_\bullet$ where each $C_n\in Pr^L_0$ is a presheaf category (but the simplicial maps are only in $Pr^L_\kappa$ where $\kappa$ is the presentability rank of $C$). For that matter, $Set^{[2]}$ is a strong generator in $Pr^L$ (i.e. the functor $Pr^L(Set^{[2]},-)$ reflects equivalences). These are weaker senses in which $Pr^L$ is "small-generated".

I'm not sure about the dual properties. Classically, $Pr^L_\kappa$ is closed in $Pr^L$ under $\kappa$-small PIE limits. And an arbitrary product of objects of $Pr^L_\kappa$ is again in $Pr^L_\kappa$, but if $(F_\alpha : C \to D_\alpha)$ is a family of functors in $Pr^L_\kappa$, the induced functor $C \to \Pi_\alpha D_\alpha$ typically is not in $Pr^L_\kappa$. So $Pr^L_\kappa$ is not closed in $Pr^L$ under small limits. It's possible that $Pr^L$ is generated by, say, $Pr^L_\omega$ under small limits -- I'm not sure. I think at least that $Pr^L_0$ is closed under small limits, though.

5. $Pr^L$ has localizations with respect to sets of morphisms.

Let $S$ be a set of morphisms, and let $\kappa$ be large enough so that all $S$ are between objects of $Pr^L_\kappa$. Then for each $\mu \geq \kappa$, we may localize $Pr^L_\mu$ at $S$ because $Pr^L_\mu$ is presentable. These localizations cohere: to see this, it suffices to check that if $X$ is local in $Pr^L_\mu$ with respect to $A \to B$, then $X$ is still local in $Pr^L_\nu$ with respect to $A \to B$ for $\nu \geq \mu$. This is not hard to see: it follows from the fact that every functor in $Pr^L_\nu$ from $A$ to $X$ or $B$ to $X$ is a colimit of functors in $Pr^L_\mu$.

6. It's not clear to me if there's a slick characterization of those localizations which arise from localizing at a set of maps.

If $Pr^L$ were presentable, these would just be the accessible localizations. But it's not.

These localizations have the property that the localization functor preserves $Pr^L_\kappa$ for sufficiently large $\kappa$, and restricts to an accessible localization on $Pr^L_\kappa$. But other localizations might have this same property -- for instance, one might localize at a saturated class of morphisms whose intersection with each $Pr^L_\kappa$ is generated by a set -- such a localization wouldn't automatically exist, but it might in some cases.

---

Here is a size estimate for $Pr^L$:

• Objects: The equivalence classes of objects are surjected onto by the following data:

  • A (small) regular cardinal $\kappa$.

  • A small, skeletal, $\kappa$-cocomplete category $C$.

    The bijection sends $(C,\kappa)$ to $Ind_\kappa(C)$. There are $\lambda$-many regular cardinals, and $\lambda$-many $\kappa$-complete small categories, for a total of $\lambda \times \lambda = \lambda$ many presentable categories up to equivalence.

• Morphisms: If $C$ is $\kappa$-cocomplete and $\mathcal D$ is presentable, then isomorphism classes of left adjoint functors $Ind_\kappa(C) \to \mathcal D$ are in bijection with the following data:

• An isomorphism class of functors $F: C \to \mathcal D$ preserving $\kappa$-small colimits.

  The bijection sends a functor $F$ to its left Kan extension. Note that $\mathcal D$ has at most $\lambda$ many isomorphism classes of objects, each of which has a small number of morphisms between them. Since $C$ is small, this bounds the number of isomorphism classes of functors by $\lambda^{|C|} = \lambda$ since $\lambda$ is inaccessible.

• 2-morphisms: Now if $F,G: \mathcal C \to \mathcal D$ are left adjoint functors, the number of natural transformations (and in particular the number of natural isomorphisms) between them is bounded by the number of objects of $\mathcal C$ times the number of morphisms between the corresponding images in $\mathcal D$, for a $\lambda$-sized sum of small sets, which has size at most $\lambda$ [2].

[1] This is actually a subtle point -- if you keep an object for each isomorphism class of presentable categories, then for each $\mathcal C \in Pr^L$, you have additional data saying how many copies of each isomorphism class of $\mathcal C$ there are. Then you have to ask what the maximum allowed number of copies is. This will depend on foundational choices like precisely how you defined the notion of presentable categories. I take the perspective that counting size the way I do above is "what you really want to do", although I suppose if you want to think about $Pr^L$ as an $\infty$-category without localizing at the equivalences, you might have some technical issues.

[2] Actually, we can do better: we can choose $\kappa$ such that $F,G$ are both Kan extended from the $\kappa$-small objects. Then any natural transformation $F \Rightarrow G$ is determined by its values on the $\kappa$-small objects. So there are actually a small number of natural transformations between $F$ and $G$. Thus when viewed as a 2-category or $(\infty,2)$-category, $Pr^L$ is large, locally large, but locally locally small. Even when viewed as a (2,1)-category or $(\infty,1)$-category, the largeness of $Pr^L$ is "not so bad": $\pi_0$ of a homspace can be large, but its connected components are small.

[3] Alexander Campbell points out that in the ordinary context, $\kappa-Cocts$ and thus the equivalent category $Pr^L_\kappa$ is not actually presentable due to strictness issues. So it's probably safest to just interpret everything I'm saying here in the $\infty$ context.

Answer 2

The following is a very long comment and works in $1$-category theory.

I claim that you can characterize very well coreflective subcategories.

My strategy works even for reflective.

There is a biequivalence of $2$-categories $$ \text{Lex}^{\circ} \cong \text{Pres}. $$

• Lex is the category of small categories with finite limits, $1$-cells are functors preserving finite limits.
• Pres is the category of finitely presentable categories, $1$-cells are accessible right adjoints.

Thus, $$\{\text{reflective subcategories of Pres}\} \leftrightarrow \{\text{coreflective subcategories of Lex}\}.$$

In a certain sense, the category Pres is the opposite of a locally presentable category. In fact, Lex is the category of algebras for the coKZ monad of free completion under finite limits (is this monad accessible?) on Cat. The category of algebras over an accessible monad defined on a locally presentable category is always locally presentable.

To conclude, as soon as one proves that Lex is locally presentable, one can derive a lot of results for Pres, just because it's the opposite of an actual locally presentable category.

Achtung!

• Pres is not the same of $Pr^L$, nor its opposite category.
• The free completion under finite limits is not a monad, it is a $2$-monad and I am not expert enough in $2$-category theory to say anything about its algebras but I strongly believe that Lex is finitely presentable as a $1$-category.
• Rosicky, Adameck and Trnkova proved that, under Vopenka Principle, every subcategory closed under colimits in coreflective in a locally presentable category. In fact this is equivalent to Vopenka. Rosicky moved the theory of cotorsion theories to locally presentable categories.