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 ?