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Let $C$ and $D$ be locally $\kappa$-presentable categories. It is written on the nLab that the category $\mathrm{Ladj}(C, D)$ of cocontinuous functors from $C$ to $D$ is again locally $\kappa$-presentable. Is this actually true? The proof of the local presentability of $\mathrm{Ladj}(C, D)$ that I know (the one used to prove Corollary 3.18 in Bird's PhD thesis) yields a rank of presentability that depends on more than just the ranks of presentability of $C$ and $D$.

However, I also was not able to come up with a counterexample to the claim. The simplest case in which things could go wrong is when $C$ is the category of presheaves on a category $A$ of cardinality at least $\kappa$. But in this case, $\mathrm{Ladj}(C, D) = [A, D]$ actually is $\kappa$-presentable. So $C$ will need to be a reflective subcategory of a presheaf category, but I was unable to construct such an example where I could compute that the rank of presentability of $\mathrm{Ladj}(C, D)$ exceeds $\kappa$.

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  • $\begingroup$ This is just to say that I tried what I thought was the simplest localization I know: let $C$ the category of sheaves on a Boolean algebra for the coherent coverage (equivalently sheaves on its Stone space, its $\omega$-presentable). In this case $Ladj(C,D)$ identifies with the category of $D$-valued cosheaves on $B$, which I assume was not going to be $\omega$-presentable. I tried with $D = Set$ and $D= Abelian group$ in both case it was $\omega$-presentable... then I tried with $D=Group$ for which I think it is not $\omega$-presentable, but I havn't been able to prove it.... $\endgroup$ May 11, 2019 at 15:35
  • $\begingroup$ I'm not sure what argument I had in mind. I may have misspoken. $\endgroup$
    – Zhen Lin
    Aug 10, 2021 at 0:38

2 Answers 2

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Let $C$ be a locally $\kappa$-presentable category. Then we may write $C = |C_\bullet|$ where each $C_n$ is a presheaf category and the geometric realization is taken in $Pr^L_\kappa$ (and so also is a geometric realization in $Pr^L$). So if $D$ is $\kappa$-presentable, then $LAdj(C,D) = LAdj(|C_\bullet|,D) = Tot LAdj(C_\bullet,D)$, where the totalization is taken in $Pr^L$ (and the cosimplicial diagram lives in $Pr^L_\kappa$ [1]). Now, $Pr^L_\kappa$ is closed in $Pr^L$ under $\kappa$-small limits if $\kappa$ is uncountable, and the simplex category is countable. So if $\kappa$ is uncountable, then $LAdj(C,D)$ is again $\kappa$-presentable. ( 1-categorically, I suppose this might even work if $\kappa = \omega$, since the cosimplicial object can be truncated at a finite stage -- but I'm not sure that $Pr^L_\omega$ is closed under finite limits in $Pr^L$.)

[1] This is actually the subtlest step, I think. By the presheaf case, $LAdj(C_n,D)$ is locally $\kappa$-presentable. The subtle part is verifying that the transition maps $LAdj(C_n,D) \to LAdj(C_m,D)$ preserve $\kappa$-presentable objects. Note that if $A$ is a small category and $D$ is locally $\kappa$-presentable, then the $\kappa$-presentable objects of $D^A$ are those functors $A \to D$ which are left Kan extended from a functor $B \to D_\kappa$ where $B$ is $\kappa$-small and $D_\kappa \subseteq D$ comprises the $\kappa$-presentable objects. The relevant functors $C_{n+1} \to C_n$ with which we are precomposing all have fully faithful right adjoints $\iota$ [2], so if $C_n \to D$ is extended from $B$, then the composite $C_{n+1} \to C_n \to D$ is extended from $\iota(B)$; thus the precomposition functor $LAdj(C_n,D) \to LAdj(C_{n+1},D)$ does indeed preserve $\kappa$-presentable objects.

[2] I suppose to be sure of this, I should say which simplicial object $C_\bullet$ I'm using. What I have in mind is the simplicial object $C_n = Ind_\kappa (F^n C_\kappa)$, where $C_\kappa \subseteq C$ is the $\kappa$-presentable objects, and $F$ is the monad which freely adjoins $\kappa$-small colimits to a category. Note that $F : Cat \to Cat$ is a monad, and that $C_\kappa$ is an algebra for $F$. Therefore, in $Cat$, the simplicial object $F^\bullet C_\kappa$ admits a split augmentation by $C_\kappa$. Moreover, the monad $F$ is lax-idempotent. Therefore, the degeneracies of the split augmented simplicial object are right adjoint right inverses to the corresponding face maps. Applying the 2-functor $Ind_\kappa$ preserves this property.

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$\mathsf{cocont}(A,B)$ is locally presentable but I don't see why it should be locally $\kappa$-presentable.

Let $A$ and $B$ be two locally $\kappa$-presentable categories.

$\mathsf{cocont}(A,B)$ is equivalent to $\kappa\mathsf{-cocont}(\text{Pres}_\kappa(A), B)$. The latter is clearly the category of models of a colimit sketch in a locally presentable category (namely $B$) and thus is locally presentable by Rem. 2.63 in Locally presentable and accessible categories by Adamek and Rosicky.

Following carefully Rem 2.59 there is no way to infer that $\kappa\mathsf{-cocont}(\text{Pres}_\kappa(A), B)$ is $\kappa$-accessible from this argument, even if the cocones have size $\kappa$.

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  • $\begingroup$ Right, this is another route to the local presentability of the internal Hom (I think it is also the one used in Higher Topos Theory). Since it passes through the theory of accessible categories I expect that it would give much worse bounds on the rank of presentability than an argument like the one in Bird's thesis. $\endgroup$ Mar 5, 2019 at 15:58
  • $\begingroup$ Yes, probably, I was just making the point that I do not believe that it is $\kappa$-accessible in general. $\endgroup$ Mar 5, 2019 at 20:53

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