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In Proposition 5.5.7.6 of Lurie's Higher Topos Theory, Lurie states that the $\infty$-category $\operatorname{Pr}^R_{\omega}$ of compactly generated $\infty$-categories and filtered-colimit-preserving right adjoints is closed under limits in $\operatorname{Cat}_\infty$. I would like to understand some detail of the proof.

Lurie says that most of the proposition follows from Theorem 5.5.3.18, which states that the inclusion $\operatorname{Pr}^R \subseteq \operatorname{Cat}_\infty$ is closed under limits. However, as $\operatorname{Pr}^R_{\omega}$ does not seem to be a full subcategory of $\operatorname{Pr}^R$, I don't understand why the result of 5.5.3.18 gives us that the adjunctions $F_\alpha: \mathcal{C}_\alpha \rightleftarrows \mathcal{C}: G_{\alpha}$ appearing in the proof of 5.5.7.6 correspond to morphisms in $\operatorname{Pr}^R_{\omega}$, i.e. why $F_{\alpha}$ preserves compact objects/$G_\alpha$ filtered colimits. Trying to trace back through the results leading up to 5.5.3.18, I get the impression that the fact that $G_{\alpha}$ is accessible in the proof of 5.5.3.18 indirectly relies on Proposition 5.4.7.7, which says that any right adjoint is accessible. However, this proposition uses that we may choose the regular kardinal $\kappa$ as high as we wish before showing that $G_{\alpha}$ is $\kappa$-continuous, and in particular cannot be applied if we wish to show this for $\kappa = \omega$.

Question: Why do the maps $G_{\alpha}: \mathcal{C} \to \mathcal{C}_{\alpha}$ appearing in the proof of 5.5.7.6 preserve filtered colimits?

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  • $\begingroup$ It may be worth reflecting on the analogous theorem in 1-categories (EDIT: for the $Pr^R$ case -- sorry, just realized you're mainly interested in $Pr_\omega^R$). $\endgroup$
    – Tim Campion
    Feb 6 at 16:07
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    $\begingroup$ It seems the same problem occurs: the results are stated for functors that `commute with sufficiently-filtered colimits', but not necessarily with $\omega$-filtered colimits. $\endgroup$ Feb 6 at 16:23
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    $\begingroup$ In the case of colimits in the 1-categorical version of $Pr^R$, I've asserted before that when $\kappa$ is uncountable, although for every small diagram in $Pr^R_\kappa$, the colimit cocone in $Pr^R$ is also in $Pr^R_\kappa$, it does not necessarily have the correct universal property in $Pr^R_\kappa$ when the diagram is bigger than $\kappa$ (induced functors may fail to be $\kappa$-accessible). I'm sure of even less when $\kappa = \omega$. I'm not sure how much of that translates to limits in $Pr^R_\kappa$, but I agree this could be delicate. $\endgroup$
    – Tim Campion
    Feb 6 at 16:32
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    $\begingroup$ Isn't there a result saying that if all functors in a diagram of $\infty$-categories preserve a given type of colimit, then the projections do too ? Namely, if all functors preserve $I$-indexed colimits, then they are computed pointwise in the limit ? $\endgroup$ Feb 6 at 16:39
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    $\begingroup$ Lemma 5.4.5.5 is the case of pullbacks, and for products it must be somewhere too (in fact it's fairly easy to prove for products), so that gives you all limits by 4.4.2.6/4.4.2.7. essentially $\endgroup$ Feb 6 at 16:42
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By lemma 5.4.5.5., the forgetful functor $Pr^R_\omega\to Cat_\infty$ preserves pullbacks : the projection functors in the pullback preserve filtered colimits.

It's easy to prove that the same thing holds for arbitrary products ( a colimit in $\prod_\alpha C_\alpha$ is computed pointwise, which means precisely that each projection preserves it), and so by 4.4.2.7, the forgetful functor preserves all limits.

The main point being the following :

Given a diagram $F: I\to Cat_\infty$ where each $F(i)$ has $K$-indexed colimits and each $i\to j$ induces a $K$-indexed colimit preserving functor $F(i)\to F(j)$, then $\lim_{i\in I}F(i)$ has $K$-indexed colimits and each projection preserves them.

You prove this the same way, using 5.4.5.5. and 4.4.2.7 to the category of $\infty$-categories that have $K$-indexed colimits, and $K$-indexed colimit-preserving functors.

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