Let $Pres$ denote the 2category of locally presentable categories (categories that are accessible/cocomplete), cocontinuous functors, and natural transformations. Let $Cat$ denote the 2category of (small) categories, functors, and natural transformations. There is an inclusion functor $i:Pres\hookrightarrow Cat$. Does this admit a left/right adjoint?

2$\begingroup$ The presheaf category is a left adjoint, that is $\mathrm{Fun}^L(P(C),D)\cong \mathrm{Fun}(C,D)$. $\endgroup$– Denis NardinAug 26, 2016 at 13:15

$\begingroup$ @DenisNardin I think there may be some size issues in trying to apply that here... $\endgroup$– David Roberts ♦Aug 26, 2016 at 13:23

1$\begingroup$ @DavidRoberts I think that the size issues appear already. when you try to include presentable categories inside small categories (that is in the definition of the functor $i$). $\endgroup$– Denis NardinAug 26, 2016 at 13:26

3$\begingroup$ Yeah, this isn't making much sense: small locally presentable categories must be preorders: a smallcocomplete small category is a preorder. $\endgroup$– Todd Trimble ♦Aug 26, 2016 at 13:44

$\begingroup$ What happens if the smallness condition is dropped? $\endgroup$– user84563Aug 26, 2016 at 17:26
1 Answer
Size conditions are not such a big deal when it comes to formulating the question, at least. We just let Cat be the nonlocallysmall 2category of locallysmall categories (perhaps we require that the object sets be no bigger than the universe, it turns out not to matter). The inclusion $\mathsf{Pres}^L \to \mathsf{Cat}$ preserves reasonable 2limits, but not 2colimits, see here. So it might have a left biadjoint but it certainly does not have a right biadjoint.
But we can show that size issues do prevent the left biadjoint from existing. Let $C$ be a discrete category with object set $C_0$ of the size of the universe. Then if $FC$ were a reflection of $C$ in $\mathsf{Pres}^L$, we would have $\mathsf{Pres}^L(FC,D) = \mathsf{Cat}(C,D) = D^{C_0}$ for any locally presentable $D$. If $D = \mathsf{Set}$, say (the category of small sets), it's easy to see that $D^{C_0}$ has the cardinality of the powerset of the universe. But the homcategory $\mathsf{Pres}^L(E,D)$ between any two locally presentable categories is no bigger than the size of the universe (this is because any cocontinuous functor $E \to D$ is the left Kan extension of a functor defined on the small subcategory of $\lambda$presentable objects for some $\lambda$; there are at most universemany such functors, and taking the union over $\lambda$ still yields universemany). So no such $FC$ can exist.
Along the lines of Dennis Nardin's comment, though, the inclusion of locally classpresentable categories into $\mathsf{Cat}$ has a left biadjoint given by sending $C$ to the free cocompletion of $C$ (which is the full category of presheaves $PC$ on $C$ when $C$ is small, but a proper subcategory of $PC$ when $C$ is large).

$\begingroup$ Are locally classpresentable categories complete? Because the small presheaves cocompletion need not be complete. $\endgroup$– Todd Trimble ♦Aug 26, 2016 at 22:01

$\begingroup$ Oh, you're right, I think they're complete by definition. I should say "cocomplete classaccessible categories". $\endgroup$– Tim Campion ♦Aug 26, 2016 at 22:17

$\begingroup$ I'm still maybe confused: a smallcocomplete accessible category is automatically smallcomplete, so the question is whether something like that carries over to this classaccessible setting (which I have not studied). $\endgroup$– Todd Trimble ♦Aug 26, 2016 at 23:15

$\begingroup$ I think it's not automatic in the class accessible case, so chorny and rosicky simply define a locally class presentable category to be a class accessible category which is both complete and cocomplete. $\endgroup$– Tim Campion ♦Aug 26, 2016 at 23:18