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I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those I need by hand, but that got me curious about what is known in general about existence of Left Bousfield localization of combinatorial model category at a set of maps:

  • Is there any known example of a combinatorial model category with a set of maps such that the left Bousfield localization does not exist ? (Edit: this is answered there)

  • Is there other kind of assumption under which we have a general theorem of existence ? For example I have the impression that putting condition on the set of maps we localize at (like localizing at a set of cofibrations between cofibration objects) might remove the need for left properness.

  • In On left and right model categories and left and right Bousfield localizations C.Barwick claims (4.13) that if we only want to construct a left semi-model category then the left Bousfield localization always exists. But he does not prove it. What is the status of this claim ? is it proved somewhere ?

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  • $\begingroup$ This comment does not answer none of your questions. But in your case, you could use a representation existence theorem, the presheaf model category would be left proper, then localize and try to transport the left Bousfield localization along the left adjoint. $\endgroup$ – Philippe Gaucher May 23 '18 at 10:03
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I have an unpublished note that proves Barwick's claim. Aspects of this story have appeared in some papers of mine with Michael Batanin, including one we published in the proceedings of the 2015 CRM conference in Barcelona on "Interactions between representation theory, algebraic topology, and commutative algebra." I'm working to make the note into a real paper. I can share a draft at some point if you want.

Specifically, what I can prove is that, if $M$ is a combinatorial semi-model category (i.e. a locally presentable category with a cofibrantly generated semi-model structure), and $C$ is a set of morphisms, then the left Bousfield localization $L_CM$ with the classes of maps in Hirschhorn's book, has a combinatorial semi-model structure and satisfies the universal property of localization (that any left Quillen functor of semi-model categories $F: M\to D$ taking $C$ to weak equivalences factors through $L_CM$). By the way, this was certainly known to Hirschhorn, as you can see from the comments in Moduli problems for structured ring spectra, by Goerss and Hopkins. Also, I think Cisinski probably knew this.

Regarding your other questions...I am pretty certain that it is not true that if you replace the maps by cofibrations between cofibrant objects, then you can get a full model structure without left properness. But I don't know an example offhand. And, I don't know an example for your first question, but of course this would be a phenomenon only visible on the model category level because every combinatorial (semi-)model category is Quillen equivalent to a left proper one, basically by Giraud's theorem.

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  • $\begingroup$ Good ! I'll indeed be interested to see your proof whenever it is ready to be shared. $\endgroup$ – Simon Henry May 23 '18 at 13:14
  • $\begingroup$ The current draft is 20 pages. I'm hoping to use this summer to get something in order that's ready to share. $\endgroup$ – David White May 23 '18 at 13:15

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