Let $Pres$ denote the 2-category of locally presentable categories (categories that are accessible/cocomplete), cocontinuous functors, and natural transformations. Let $Cat$ denote the 2-category of (small) categories, functors, and natural transformations. There is an inclusion functor $i:Pres\hookrightarrow Cat$. Does this admit a left/right adjoint?
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2$\begingroup$ The presheaf category is a left adjoint, that is $\mathrm{Fun}^L(P(C),D)\cong \mathrm{Fun}(C,D)$. $\endgroup$ – Denis Nardin Aug 26 '16 at 13:15
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$\begingroup$ @DenisNardin I think there may be some size issues in trying to apply that here... $\endgroup$ – David Roberts Aug 26 '16 at 13:23
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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 Nardin Aug 26 '16 at 13:26
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3$\begingroup$ Yeah, this isn't making much sense: small locally presentable categories must be preorders: a small-cocomplete small category is a preorder. $\endgroup$ – Todd Trimble♦ Aug 26 '16 at 13:44
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$\begingroup$ What happens if the smallness condition is dropped? $\endgroup$ – user84563 Aug 26 '16 at 17:26
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Size conditions are not such a big deal when it comes to formulating the question, at least. We just let Cat be the non-locally-small 2-category of locally-small 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 2-limits, but not 2-colimits, 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 hom-category $\mathsf{Pres}^L(E,D)$ 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 universe-many such functors, and taking the union over $\lambda$ still yields universe-many). So no such $FC$ can exist.
Along the lines of Dennis Nardin's comment, though, the inclusion of locally class-presentable 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).
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$\begingroup$ Are locally class-presentable categories complete? Because the small presheaves cocompletion need not be complete. $\endgroup$ – Todd Trimble♦ Aug 26 '16 at 22:01
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$\begingroup$ Oh, you're right, I think they're complete by definition. I should say "cocomplete class-accessible categories". $\endgroup$ – Tim Campion Aug 26 '16 at 22:17
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$\begingroup$ I'm still maybe confused: a small-cocomplete accessible category is automatically small-complete, so the question is whether something like that carries over to this class-accessible setting (which I have not studied). $\endgroup$ – Todd Trimble♦ Aug 26 '16 at 23:15
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$\begingroup$ 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 '16 at 23:18
