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finished typing part 1) and 2) of the proof of the existence theorem at Bousfield localization of model categories
I completed the existece proof with the bit showing the accessibility of the S-local weak equivalences.
Then I added a section prerequisite for the proof that lists briefly all the material at other entries, previously just having been linked to, that enters the proof.
Based on some feedback that I got behind the scenes, I have now rearranged the material, to make it more light-weight at the beginning.
the section Properties now has the central properties one can deduce for a Bousfield localization that do not require combinatorial model category technology
the section Existence of ... has the existence result for the combinatorial case pretty much as before, but minus the material that has been moved to "Properties"
and the Definition section is now such that it downplays the "derived hom" and instead tends to arrange things such that they are properly cofibrant or fibrant . Because it was suggested to me that invoking the derived hom concept makes this look, to the non-expert reader- much more arcane than it is.
I just hope that the interconnecting logic between the paragraphs is now still consistent. I fddled a bit with adjusting the statements to the slightly changed assimptions.
added the statement that left Bousfield localization is functorial with respect to Quillen equivalences, here.
I have added statement and proof (here) that given a left Bousfield localization, then local fibrations between local objects are global fibrations.
(I am assuming functorial factorization, which is of course provided under the generic assumption of the entry that we are localizing a cofibrantly generated model category. But probably one can get the statement more generally.)
I have made more explicit statement and proof that functorial fibrant replacement in a left Bousfield localized model category is a model for the derived adjunction unit, here
added pointer to:
(Answered #8 in the other thread.)
Great. Thanks for looking into it.
But let’s add references for the claims. On the non-simplicial combinatorial case, is there a reference beyond Barwick’s unpublished preprint?
is there a reference beyond Barwick’s unpublished preprint?
Barwick’s paper has been published many years ago. I already added a reference in the previous revision.
I see. Thanks!
I have made explicit (here) that the published version of Barwick’s article has a different title than the arXiv version.
In fact, also the Theorem-numbers are different, and I have now tried to adjust this in the text where they are being referenced.
[edit: this may need further attention. But I am in a rush now to get to an airport…]
Finally, I deleted the now duplicated previous item listing just the arXiv version.
added pointer to:
Re #16:
Barwick’s paper “On left and right…” is a fusion of his three arXiv preprints from August 2007:
Section 3: On Reedy Model Categories
Section 4: On (Enriched) Left Bousfield Localization of Model Categories
Section 5: On the Dreaded Right Bousfield Localization
Currently, the article also cites the last paper separately, but it is incorporated as Section 5.
added pointer to:
For completeness I made a References-subsection “Monoidal case” (here), currwntly just duplicating the references earlier added at monoidal localiztion
Re #21: After Barwick, the earliest reference for monoidality of left Bousfield localizations is https://arxiv.org/abs/0907.0730v4, Lemma 28.
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