So I am trying to learn a bit about dg categories from Toën's notes, "Lectures on dg-categories" http://www.math.univ-toulouse.fr/~toen/swisk.pdf and in particular I would like to understand proposition 11 on page 52. The reference he cites is around 300 pages and seems to develop a lot more than maybe necessary for this particular statement. I'm happy for the moment to forget about perfect complexes and just try the statement for bounded quasicoherent complexes, although I'd definitely be interested if people wanted to discuss that case too.
Maybe I am talking absolute nonsense, but my vague impression is that the basic statement should be that homotopy pullback is a derived functor of the categorical pullback with respect to localization of the category of dg categories by inverting weak equivalences and that the dg-category $L(X)$ that Toen is discussing is an acyclic replacement for $C(O_X)$ with respect to this functor. Can anyone give clean references for this sort of theorem? Though I appreciate that this theorem probably lies in the realm of $\infty$ categories, I think I'd prefer discussion less fancy and closer in appearance to the above theorem if possible. Ideally, it should be possible to walk me through this theorem by hand? Presumably this all comes down at the end of the day to some relatively simple homological algebra, but my head isn't clear enough with respect to this stuff to work it out.