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Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For each regular cardinal $\kappa$, let $Pr^L_\kappa$ be the non-full subcategory of locally $\kappa$-presentable categories and left adjoint functors which preserve $\kappa$-presentable objects (equivalently, which have $\kappa$-accessible right adjoints).

I believe that limits (PIE limits in the ordinary setting) in $Pr^L$ are computed at the level of the underlying category, and that $Pr^L_\kappa$ is closed in $Pr^L$ under $\kappa$-small limits (at least for $\kappa > \omega$). Moreover, an arbitrary product of objects of $Pr^L_\kappa$ is again in $Pr^L_\kappa$ and the projection functors are even in $Pr^L_\kappa$. However, if $(F_\alpha : C \to D_\alpha)_\alpha$ is a family of functors in $Pr^L_\kappa$, the induced functor $F: C \to \Pi_\alpha D_\alpha$ is not typically in $Pr^L_\kappa$ when the product is $\kappa$-sized or larger.

Thus $Pr^L_\kappa$ is not closed in $Pr^L$ under small limits. This raises the following

Question: Is $Pr^L$ generated under small limits by $Pr^L_\kappa$ for some $\kappa$? How about $\kappa = \omega$?

Notes:

  • I believe that $Pr^L_0$ (the category of preseheaf categories) is closed under limits in $Pr^L$, so I think the smallest candidate $\kappa$ in the above question is $\kappa = \omega$. This is what I meant in the question title, where "finitary left adjoint" is meant to indicate "morphism in $Pr^L_\omega$".

  • Exactly what it means to be generated under limits by a non-full subcategory is a bit unclear. But let's at least stipulate that if every object of $Pr^L$ is a limit of a diagram in $Pr^L_\kappa$, then the answer to the question is yes. I suppose I don't even know, for any $\kappa$, if the full subcategory of $Pr^L$ on the objects of $Pr^L_\kappa$ generates $Pr^L$ under limits in the usual sense, so figuring that out would be a good start.

    EDIT: Another consequence of an affirmative answer to the question (which might be easier to think about) would be the following: every presentable category would be coreflective in a $\kappa$-presentable category for the $\kappa$ in the question. This can't happen with $\kappa = 0$, but maybe it can happen for $\kappa = \omega$.

  • EDIT: Here's a data point. Let $CMet \in Pr^L_{\omega_1}$ be the category of complete metric spaces and contractive maps. Let $PMet \in Pr^L_\omega$ be the category of pseudometric spaces and contractive maps. For each $\delta \geq 0$ there is a functor $\delta_\ast: PMet \to PMet$ preserving underlying sets with $d_{\delta_\ast X}(x,y) = min(d_X(x,y) - \delta, 0)$. These form an inverse system for $\delta > 0$, and I believe the limit of this inverse system is none other than $CMet$.

  • For context, this question arose from reflecting on KotelKanim's question.

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  • $\begingroup$ I’d recommend having a look at arxiv.org/abs/1409.5934. I don’t know that it has exactly the answer to your question, but it has a lot of interesting results about the relationship between the $\mathrm{Pr}_\kappa^L$. $\endgroup$ – Noah Snyder Aug 5 '18 at 13:27

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