# Is the canonical model structure on strict $\infty$-Cat left proper?

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).

• 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? – Tim Campion Apr 10 at 14:53
• @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 ? ) – Simon Henry Apr 10 at 14:57
• @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. – Simon Henry Apr 10 at 15:13
• 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. – Tim Campion Apr 10 at 15:16
• 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. – Tim Campion Apr 10 at 15:32

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