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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→(X↓C) be a κ-good S-tree rooted at X for a collection of morphisms S in C, where κ 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 κ-small downward-closed B⊆A, the colimit of the restricted diagram, varinjlimD|B is κ-compact in (X↓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 κ-compact if hX(−):=Hom(X,−) preserves all κ-filtered colimits (where κ-filtered means “< κ“-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 Uα→Vα∈S.
We say that an S-tree is κ-good if for all of the morphisms Uα→Vα above, Uα and Vα are κ-compact, and such that for any α∈A, the subset {β:β < α}⊆A is κ-small.
Edit: It’s easy to reduce the proof to showing that D(α) is κ-compact, since projective limits of diagrams B→Set are |Arr(B)|-accessible (and therefore κ-accessible since B is κ-small), we perform the computation for I a κ-filtered poset, and F:I→C, assuming that D(α) is κ-compact for all α∈B:
varinjlimIHomC(varinjlimBD,F)≅varinjlimIvarprojlimBopHomC(D,F)≅varprojlimBopvarinjlimIHomC(D,F)≅varprojlimBopHomC(D,varinjlimIF)≅HomC(varinjlimBD,varinjlimIF)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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