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Do homotopy colimits commute with homotopy colimits? The setting I am thinking of is that of a triangulated category with a model, but it would be interesting to have more general answers as well. A good reference would also be appreciated.

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Yes. See theorem 24.9 of this paper. Here's a somewhat less abstract reference.

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    $\begingroup$ Ah, it was a few years ago when the answer was posted, but it turns out this is exactly what I was looking for right now. Thank you! For those who want to use this in their papers, let me just leave a remark here that it was published as part of Memoir of AMS, Vol 155, No. 736 (2002). $\endgroup$
    – Jinhyun Park
    Commented Mar 2, 2012 at 9:34

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