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I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all the techniques I have ever seen for constructing full Quillen model structures (by opposition to construction of left or right semi-model structure) relies at some point on an argument that also implies either left or right properness.

A typical example of this is when a model structure is constructed by combining an argument that would usualy produce a left semi-model structure with the fact that every object is cofibrant, but this also implies left properness (that is typically what happen with Cisinki model structures, and their generalization by Olschok).

I'm very curious to know if the I above can be replaced by a we, and to be honest I would be very interested in seeing an example of a construction of model structure that use a very different kind of argument than those I already know. Hence my question:

Is there any known, naturally occurring, example of a Quillen model category that is neither left nor right proper ?

I don't doubt that model structures that are neither left nor right proper exist, it should be easy to built one along the lines of the example Reid Barton gave as an answer to this recent other question of mine, i.e. as a model structure on a partially ordered set. But I wouldn't consider a satisfying answer to my question. I'm really more interested in "natural" or "interesting" examples. Ideally a model structure that could have some interest in its own right, but at this point, maybe just an example which is not a poset would already be a good step.

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    $\begingroup$ I like this question! Maybe it's worth mentioning that a non-proper model structure can't have the same weak equivalences as a proper one. (See the comments here by Rezk, Nikolaus, and Shulman.) So, for example, an answer to this question can't come from shrinking the class of cofibrations of a Cisinski model structure so that not everything is cofibrant; it seems like it really would require a fundamentally different approach, like you're saying. $\endgroup$
    – Matt Feller
    Commented Aug 24, 2019 at 15:55
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    $\begingroup$ So to list some general techniques for constructing model structures: Bousfield localization assumes properness, while mixed and intermediate model structures have the same weak equivalences as some other model structure and so don't take us out of the world of properness. What about left and right lifting? E.g. if we right-lift a model structure that is not right proper, could we obtain one that is neither right nor left proper? $\endgroup$
    – Mike Shulman
    Commented Aug 25, 2019 at 13:39
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    $\begingroup$ @MikeShulman : Indeed, Transfer right transfer preserves right properness, and left transfer preserves left properness, but they can potentially destroy the other properness assumption. But I what I mean in this case is that I do not know any examples where one can prove that the transfered model structure exists (and is really a Quillen model structure) without using some technique that also implies or use right or left properness. $\endgroup$
    – Simon Henry
    Commented Aug 25, 2019 at 16:52
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    $\begingroup$ @SimonHenry I was just suggesting what seemed to me the most promising place to look for examples. But I see your point. $\endgroup$
    – Mike Shulman
    Commented Aug 26, 2019 at 4:17
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    $\begingroup$ I think Inna Zakharevich has some unusual ways to construct model structures. If I recall correctly, one way has something to do with starting with one model structure and then passing to an elementarily equivalent category to get a model structure on it. I think she also has studied model structures on posets in some detail, and may have interesting ways of constructing model structures on them. But I'm not sure such methods can break properness. $\endgroup$
    – Tim Campion
    Commented Aug 26, 2019 at 17:05

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