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At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which were not left proper, but which did admit Bousfield localization. There are some trivial examples, but I'm interested now to hear if anyone knows any interesting or surprising examples in this vein. I do have one example (from the person who asked the question), but it's from unpublished work so I'm not going to write about it here. Since localization with respect to a set of maps is the same as localization with respect to their coproduct, I'll just focus on localization with respect to a single map $f$.

Are there interesting or surprising examples of model categories which are not left proper but which admit left Bousfield localization with respect to every map $f$? What about if we restrict to some non-trivial class of maps $f$?

Also, this question got me thinking of whether one can do Bousfield localization without assuming the model category $M$ is cellular or combinatorial. It has always seemed to me that these hypotheses are there just to make the proofs work (cellularity for Bousfield's original proof and combinatoriality to simplify things even further). Because the proof that the localized model structure exists (i.e. that it is a model category) always comes down to the Transfer Principle for transferring structures across an adjunction, so there is always some drawn out analysis of pushouts. Perhaps because of this there is no hope of removing the hypotheses, but I still want to ask to see if that's what the experts think.

Are there interesting or surprising examples of model categories which are neither cellular nor combinatorial but which admit left Bousfield localization with respect to every map $f$? Or has someone studied this question and found subclasses of maps $f$ which work?

Most importantly:

Has anyone tried to classify model categories which admit Bousfield localization? Are there any theorems of the form “if $M$ admits Bousfield localization with respect to a set of maps $S$ then some other thing holds” in the literature?

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David, my feeling is that up to set theory any model category is Quillen equivalent to one which has all Bousfield localizations etc. –  Fernando Muro Oct 29 '12 at 22:03
    
@Fernando this was my thought too originally, and I was hoping some of Dan Dugger's early work might do this. I was especially interested in Replacing Model Categories by Simplicial Ones. Unfortunately for me, his theorems only work if $M$ is left proper and combinatorial or left proper and cellular. Do you have any reference where I could read more to try and make your answer work? Perhaps something by Cisinski? –  David White Oct 29 '12 at 22:47
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@Fernando: I doubt this is true. Consider any model category in which the weak equivalences are isomorphisms. The associated $(\infty,1)$-category has discrete mapping spaces, but the localized $(\infty,1)$-category at some map typically has nondiscrete mapping spaces, so you cannot find it as a full subcategory of the original $(\infty,1)$-category (which you could if it was a Bousfield localization in any Quillen equivalent model category). –  Marc Hoyois Oct 29 '12 at 23:31
    
@Marc, if weak equivalences are isomorphisms then left properness is automatic. How would this fit into your heuristics? –  Fernando Muro Oct 30 '12 at 10:16
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@Fernando: Marc's example is left proper, as you say, but in general it won't be cellular, combinatorial, or even cofibrantly generated. On the other hand, I don't think I know offhand of an example of a complete and cocomplete category whose localization at some map provably has nondiscrete mapping spaces. Apparently, such a category can't be both locally presentable and have its trivial model structure cofibrantly generated. An interesting question... –  Mike Shulman Nov 6 '12 at 16:30

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One relevant reference, though not answering any of the original questions, is a paper by George Raptis On the cofibrant generation of model categories, where it is shown that under Vopenka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial model category. Jiry Rosicky proves the converse direction in Are all cofibrantly generated model categories combinatorial?, so this question is set-theoretical.

An example for your second question may be found in the article by Emmanuel Dror Farjoun Homotopy theories for diagrams of spaces, where a non-cofibrantly generated model category is constructed, which I learned to localize in my theses. It admits functorial localizations with respect to any set of maps and even with respect to some particularly nice classes of maps. I do not have a general theory of left Bousfield localizations for such categories, but it can be often constructed from the functorial localization by Bousfield-Friedlander technique.

By the way, the localization with respect to a set of maps may not be expressed as a localization with respect to a single map in a general model category. See our paper with Carles Casacuberta The orthogonal subcategory problem in homotopy theory for a simple counterexample.

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Nice answer, and nice to see you here! –  Fernando Muro Feb 10 '13 at 19:11
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Thanks, and welcome to MathOverflow! I was unaware of the paper in your second paragraph...I'll have to look into it. I'm a fan of the papers in both your first and third paragraphs, and I hope they get as much publicity and interest as possible. –  David White Feb 10 '13 at 21:12
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@Fernando: Thanks! I held myself back for quite a long time, but the temptation to join had finally overcome. @David: I have a comment on your third question. I do not know many examples, where the localization does not exist. In fact, the only example that comes to mind (apparently, I learned it from Bill Dwyer) is the following: Consider the category of pointed simplicial sets with the trivial model structure, then the localization with respect to the class of all weak equivalences does not exist, since homotopy is not concrete. –  Boris Chorny Feb 11 '13 at 13:33
    
Previous comment continued. Before we have a bunch of counterexamples of different nature we have no chance to classify model categories which admit Bousfield localizations. –  Boris Chorny Feb 11 '13 at 13:36

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