I posted this as a question over at MO, but I figured I’d post it here as well:
Question:
Let be a -good -tree rooted at for a collection of morphisms in , where 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 -small downward-closed , the colimit of the restricted diagram, is -compact in .
Why is this true? (It is stated without proof.)
Definitions:
For your convenience, here are the definitions:
Recall that an object in is called -compact if preserves all -filtered colimits (where -filtered means “< “-filtered, since the terminology is different depending on the source).
Recall that an -tree rooted at for a collection of morphisms in consists of the following data:
- An object in C (the root)
- A partially ordered set whose order structure is well-founded (the index)
- A diagram such that given any element , the canonical map
is the pushout of some map .
We say that an -tree is -good if for all of the morphisms above, and are -compact, and such that for any , the subset < is -small.
Edit: It’s easy to reduce the proof to showing that is -compact, since projective limits of diagrams are -accessible (and therefore -accessible since is -small), we perform the computation for a -filtered poset, and , assuming that is -compact for all :
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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