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Is there a notion of Reedy cofibration category written down somewhere ? I don't want to reinvent the wheel.

Motivation: I ask the question because I have a cofibration category in the sense of https://ncatlab.org/nlab/show/cofibration+category and I need to consider the diagrams in this cofibration category over the direct Reedy category $0\to 1 \to 2 \to \dots$. And I need the fact that the colimit functor is a left Quillen adjoint somehow. Does it make sense for a cofibration category ?

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    $\begingroup$ I’m pretty sure this is done; off the top of my head I don’t remember where, but it should be mentioned in the bibliography of my paper with Chris Kapulkin Homotopical inverse diagrams in CwA’s. Possibly it’s in work of Kapulkin and/or Szumilo? I don’t have time to search it up now, but can try later if someone else doesn’t find a good source sooner! $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Oct 8 at 10:44
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    $\begingroup$ @PeterLeFanuLumsdaine Indeed, it's Theorem 9.2.4 of Cofibrations in Homotopy Theory. $\endgroup$
    – Philippe Gaucher
    Commented Oct 8 at 12:39

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