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I am reading Lurie's notes on Bousfield localization, attached below. I have the following questions:

  1. I don't get why the inclusion $C \subset Sp$ preserves homotopy colimits from assumption (*)

  2. Can I have the precisely statement and reference to the version of the adjoint functor theorem?

  3. Does everything work if I replace $Sp$ by $D(A),D^+(A), D^-(A)$ where $A$ is an abelian category?

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  1. It's not from assumption (*), but from the assumption about that that $C$ is closed under homotopy colimits.

  2. I guess he's talking about Brown representability. Just Google that, noting that the homotopy categories of $C$ and of spectra are triangulated.

  3. Yes, although chain complexes is a better analog of spectra than the derived category is.