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  • $\begingroup$ @DavidBen-Zvi: Really interesting answer, btw I'm a little confused here by "models" do you mean "model categories"? $\endgroup$ Commented Dec 28, 2011 at 13:32
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    $\begingroup$ Yes - it seems to me that when you need to identify a particular differential or explicit relation or generally any argument that doesn't follow for purely structural reasons, you will need to get behind the gleaming facade and into the inner workings of the machine, which typically means working with model categories. This is analogous to needing to choose a basis or coordinates to do most explicit calculations in the "old world", unless you can obviate them by abstract nonsense. $\endgroup$ Commented Dec 28, 2011 at 16:55
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    $\begingroup$ I think this is a great point--one great thing about becoming familiar with oo-categories is that the kinds of "formal" arguments from classical category theory will carry over, really easily, into the oo-category world. To a listener it might be frustrating if Person A says the word "adjunction" instead of saying "oo-adjunction;" but one reason Person A can feel such license is because, really, so much of classical terminology and intuition can be transported into this framework. $\endgroup$ Commented Dec 28, 2011 at 18:28