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expanded the Idea-section at Bousfield localization (by shamelessly following the introduction of Clark Barwick's nice article, referenced there)
added the Smith-Barwick existence theorem to Bousfield localization
expanded the "Properties" section at Bousfield localization with more and more detailed theorems
Zoran, I don't know. by and I suppose you mean the model categories of complexes insteadof the derived categories? Otherwise I am not sure what Bousfield localization should mean, for derived categories.
At Existence of Bousfield localization I spelled out the proof (from Barwick's article) that every left proper combinatorial model category has all Bousfield localizaitons at sets of morphisms. Using Jeff Smith's recognition theorem for combinatorial model categories, this is comparatively easy.
But I seem to have one confusion about filtered colimits:
Question
Why is every -filtered colimit in a model category necessarily a homotopy colimit, for sufficiently large ?
I seem to have a worse confusion, which probably shows that I am suffering from an elementary misundersstanding of something:
at small object it says that that a -compact obect for regular is also compact for all greater than . But clearly also for all lower than . So for all ? What's wrong here??
Added also Dugger's theorem saying that every combinatorial model category is a left Bousfield localization of the global projective model structure on simplicial presheaves on some small category.
Do you have a reference for that?
Thanks, Zoran. Currently I will concentrate on the Bousfield localization of model categories. I hope to find time to look into localization of triangulated categorries later. I find it a bit confusing that this is supposed to be called Bousfield localization, too. All the literature that I have looked at so far (Hirschhorn, Lurie, Barwick) use "Bousfield localization" for localization of model categories and don't mention any other flavor. I am hoping the one on triangulated categories is the image under passing to homotopy categories of the full one applied to categories of complexes.
I am planning to type into the nLab entry a full detailed list of statements and proofs for Bousfield localizations of combinatorial model categories. I was thinking a bit whether to follow Hirschhorn, Lurie or Barwick on this, which are all similar but differ a bit in the approaches used in the proofs. Barwick's proof looks very slick, but so far I failed to follow one of his central lemmas:
he says that in a left proper model category all pushouts of cofibrations are homotopy pushouts. I do see the weaker statement that pushouts along cofibrations with cofibrant domain are homotopy pushouts. I have typed this now into Proper model category -- propeties. But presently I fail to follow Barwick's proof of the stronger statement on page 9 here.
added section relation to presentable (oo,1)-categories -- mainly to serve as a reply for a Mathverflow question for the moment
rewrote the Idea section of Bousfield localization, trying to make it more succinct.
more details in the proof of the main theorem here, but am too tired now, will continue tomorrow
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