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I posted this as a question over at MO, but I figured I’d post it here as well:
Let $D:A\to (X\downarrow C)$ be a $\kappa$-good $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$, where $\kappa$ is a fixed uncountable regular cardinal. Then according to the proof of Lemma A.1.5.8 of Higher Topos Theory by Lurie, for any $\kappa$-small downward-closed $B\subseteq A$, the colimit of the restricted diagram, $\varinjlim D|_B$ is $\kappa$-compact in $(X\downarrow C)$.
Why is this true? (It is stated without proof.)
For your convenience, here are the definitions:
Recall that an object $X$ in $C$ is called $\kappa$-compact if $h^X (-) := Hom(X,-)$ preserves all $\kappa$-filtered colimits (where $\kappa$-filtered means “< $\kappa$“-filtered, since the terminology is different depending on the source).
Recall that an $S$-tree rooted at $X$ for a collection of morphisms $S$ in $C$ consists of the following data:
is the pushout of some map $U_\alpha\to V_\alpha\in S$.
We say that an $S$-tree is $\kappa$-good if for all of the morphisms $U_\alpha\to V_\alpha$ above, $U_\alpha$ and $V_\alpha$ are $\kappa$-compact, and such that for any $\alpha\in A$, the subset $\{\beta: \beta$ < $\alpha \}\subseteq A$ is $\kappa$-small.
Edit: It’s easy to reduce the proof to showing that $D(\alpha)$ is $\kappa$-compact, since projective limits of diagrams $B\to Set$ are $|Arr(B)|$-accessible (and therefore $\kappa$-accessible since $B$ is $\kappa$-small), we perform the computation for $I$ a $\kappa$-filtered poset, and $F:I\to C$, assuming that $D(\alpha)$ is $\kappa$-compact for all $\alpha\in B$:
$\begin{matrix} \varinjlim_I Hom_C(\varinjlim_B D, F)&\cong& \varinjlim_I\varprojlim_{B^{op}} Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}} \varinjlim_I Hom_C(D,F)\\ &\cong& \varprojlim_{B^{op}} Hom_C(D,\varinjlim_IF)\\ &\cong& Hom_C(\varinjlim_B D,\varinjlim_IF) \end{matrix}$Edit 2: I think the above reduction actually won’t work, since it doesn’t use the hypothesis that B is downward-closed.
nForum Edit 1: For some reason, \varinjlim, \varprojlim, and \cdot don’t render.
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