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Is the canonical (or Folk) model structure on the category of (strict) $\infty$-categories as constructed by Lafont, Métayer and Worytkiewicz in A folk model structure on omega-cat left proper ?

All its objects are fibrant, so it is right proper, but left properness is not clear at all. I tend to think it is not, but the pushout of a cofibration in $\infty$-Cat are still a relatively well-behaved operation so it might very well be. In any case, I don't know explicit examples of pushouts showing it is not left proper.

I would also be interested in answers to this question for the folk model structure for 2 or 3-categories (either strict or semi-strict) as this would very probably shed some light on the question.

Motivation: I've stumbled on this for several reasons, but one of them is that the folk model structure has many interesting left Bousfield localizations. For example for each $n$ it can be Bousfield localized so that the fibrant objects of the localization are the $(\infty,n)$-category in the sense that all arrows of dimension $>n$ are weakly invertible (contrary to Ara–Métayer's "generalized Brown-Golanski" model structure in The Brown-Golasinski model structure on strict $\infty$-groupoids revisited, where the arrows of dimension $>n$ are strictly invertible; whether these two things are Quillen equivalent is as far as I know an open problem).

But in order for the localization to be an actual Quillen model category (instead of a semi-model category) one need left properness.

I'm very curious to know if this localization is an example of a Bousfield localisation of non proper model structure (because those have surprising properties and I don't know really natural examples of this so far), or if it is actually a Quillen model structure (which would also be nice).

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  • $\begingroup$ The Street nerve is a fully faithful functor $Cat_\omega \hookrightarrow Strat$ from strict omega-categories to stratified simplicial sets, which has a left adjoint exhibiting it as a localization. I suspect the canonical model structure on $Cat_\omega$ is projectively-transferred along this adjunction from the Verity model structure for complicial sets. Verity's model structure is left proper, so this should imply that the canonical model structure is also left proper, right? $\endgroup$
    – Tim Campion
    Commented Apr 10, 2021 at 14:53
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    $\begingroup$ @TimCampion Assuming it is true that the model structure is transfered from Verity's model structure (because that does sounds reasonable), I don't think this implies that left properness is preserved does it ? I think for example any tractable simplicial model structure is left transfered from the Ching-Riehl model structure on its coalgebraic cofibrant object which is always left proper. (or maybe the "localization" part is important in your argument ? ) $\endgroup$
    – Simon Henry
    Commented Apr 10, 2021 at 14:57
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    $\begingroup$ @TimCampion : Ah I understand your argument now, indeed the localization part was important. the problem is in the left transfered: the left adjoint isn't a Quillen functor. In the verity model structure you have a cofibration that "makes a cell marked", which is send to the map in $Cat_\omega$ that collaps a cell to an identity, which isn't a cofibration. $\endgroup$
    – Simon Henry
    Commented Apr 10, 2021 at 15:13
  • $\begingroup$ If the argument makes sense to you, then you're ahead of me -- your first objection is compelling to me! One bit of wiggle room that could be used is the fact that left properness is a property that depends only on the weak equivalences and not the cofibrations. So if the transferred model structure exists, is left proper, and has the same weak equivalences as the canonical model structure, that implies the canonical model structure is also left proper. $\endgroup$
    – Tim Campion
    Commented Apr 10, 2021 at 15:16
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    $\begingroup$ I think maybe the transferred model structure does not exist, though. I think there are strict $\omega$-categories $C$ not weakly equivalent to a strict $\omega$-category $C'$ with fibrant Street nerve. $\endgroup$
    – Tim Campion
    Commented Apr 10, 2021 at 15:32

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The canonical model structure on 2-Cat is left proper. This is proven in Steve Lack's original paper A Quillen model structure for 2-categories that constructs this model category. The proof involves a long calculation of the pushout of trivial fibrations along the generating cofibrations.

(Side note: many of us believe the adjective "folk" is not really appropriate for these model structures, except possibly in the case $n=1$.)

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  • $\begingroup$ I feel silly, I did tried to look at it, but I looked at the wrong paper ^^ (I only looked at the one correcting a mistake in this one, but that does not mention properness) Thanks for the pointer $\endgroup$
    – Simon Henry
    Commented Apr 10, 2021 at 14:51
  • $\begingroup$ By the way, what name would you suggest to replace folk by ? I agree it isn't a great name, but I haven't seen any better suggestion. $\endgroup$
    – Simon Henry
    Commented Apr 10, 2021 at 14:52
  • $\begingroup$ I suspect Lack's key lemma 6.2 (pushouts by generating trivial cofibrations preserve trivial fibrations) is false in dimension >2. I think I had a counter example to this sort of claim at some point (I'll try to remind myself what it is... maybe it'll say something about the general question) $\endgroup$
    – Simon Henry
    Commented Apr 10, 2021 at 15:10
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    $\begingroup$ On the nLab we chose "canonical". It's maybe not great, but it's better than "folk". $\endgroup$
    – Mike Shulman
    Commented Apr 10, 2021 at 15:43
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    $\begingroup$ At a talk I heard someone ask Lack whether the model structure on bicategories is also left proper. He said he hadn't checked it and had no plans to, but he had no reason to think it wouldn't be. However, I could easily believe that stuff would start to go wrong in dimensions >2, although I haven't thought about it myself. $\endgroup$
    – Mike Shulman
    Commented Apr 10, 2021 at 15:45

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