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In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also hinted at in the beginning of the proof of Proposition 1.2.1.19 in Lurie's Higher Algebra. However, I have trouble to write down the details. Could someone clarify this for me please? Also, is there a reference available?

Alternatively, is there a way to see this through the Bousfield-Kan formula?

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    $\begingroup$ There is a reference in an appendix of HTT - I'll try to locate it when I find the time. But it is rather explicit in HTT $\endgroup$
    – Maxime Ramzi
    Commented Dec 25, 2023 at 9:35

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There is a reference for this in Lurie's Higher topos theory - specifically, see Theorem 4.2.4.1, Corollary 4.2.4.8.

The key to these is Proposition 4.2.4.4, which itself relies heavily on Appendix A in HTT. In some sense, Appendix A is the place for all these comparisons between model-theoretic concepts and their $\infty$-categorical analogue.

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  • $\begingroup$ This is just a sequential system and I guess that it is much easier than anything using model categories. Namely, you only have to identify countable products and equalizers (or fibers). $\endgroup$
    – Z. M
    Commented Feb 17 at 21:17
  • $\begingroup$ The question was about "homotopy limits", I guess I didn't check the Stacks project to see what special case it was. Also I thought I might as well cite the appropriate generality. @Z.M $\endgroup$
    – Maxime Ramzi
    Commented Feb 17 at 21:58

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