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Feb 5, 2020 at 14:11 comment added Ivan Di Liberti Thank you very much for these references! I'll check them! :)
Feb 5, 2020 at 14:10 comment added D.-C. Cisinski You should have said sooner. There are plenty of alternative references, such as Theorem 2.5.9 of paper arXiv:1507.01564 of Barnea, Harpaz and Horel, or the thesis of Aaron Mazel-Gee, or in my book on higher categories (if you look for the greatest level of generality).
Feb 5, 2020 at 11:25 comment added Ivan Di Liberti This is a very good point, thanks. I have just shown that its associated infinity category has the proper notion of colimits. When I asked the question, this was in fact my main concern.
Feb 5, 2020 at 11:11 comment added D.-C. Cisinski Is it reasonable to try to answer Q1 before Q2? When you say it is sufficient to deal with Reedy diagrams, you must have in mind a precise notion of homotopy colimits: which one are you considering here?
Feb 5, 2020 at 11:04 history edited Ivan Di Liberti CC BY-SA 4.0
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Feb 5, 2020 at 10:42 history edited Ivan Di Liberti CC BY-SA 4.0
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Feb 5, 2020 at 9:30 history answered Ivan Di Liberti CC BY-SA 4.0