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In his well-known paper "Disks, Duality and $\Theta$-Categories, Joyal defines at the very end a $\Theta$-Category to be a cellular set suitably "fibrant".

I was wondering whether someone has worked on this proposed model for weak $\omega$-Category, and its relationship with Dimitri Ara's Higher-QuasiCategories and Harry Gindi's attempt of putting a model structure on cellular sets.

A question could be, for instance: is it the case that the localizer for higher-quasicategories produces the same fibrant objects, namely those with the inner-horns filling property?

Any idea or comment is welcome!

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    $\begingroup$ Isn't this answered in Ara's paper arxiv.org/abs/1206.4354 ? His higher quasi-categories are almost like Joyal's definition, but have some extra "completeness" condition. $\endgroup$ – Mike Shulman Oct 13 '16 at 20:37
  • $\begingroup$ Thanks for your comment Mike. I think what one needs to prove is that the localizer generated by those two sets of maps gives as trivial cofibrations the same class as the saturation of the "inner horns inclusion", which he does in the case n=1 in Theorem 5.20 (using a well known Joyal's result). $\endgroup$ – Edoardo Lanari Oct 14 '16 at 2:55
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    $\begingroup$ I thought his point is that this is not true. $\endgroup$ – Mike Shulman Oct 14 '16 at 3:56
  • $\begingroup$ It's not clear to me, because I know that Joyal and Cisinki's initial idea doesn't work (as pointed out in Harry Gindi's work too) so maybe you get the inner horns if you just consider the "globularity" part, but then you need to add "completeness". I dont know if this interpretation is correct though. $\endgroup$ – Edoardo Lanari Oct 14 '16 at 4:12
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I think my preprint answers this question in the appendices when I compare the horizontal Joyal model structure on $\Theta[C]$ sets with the Rezk model model structure for $\operatorname{Se}_C\cup \operatorname{Cpt}_C$-local Reedy-fibrant simplicial presheaves.

If you unwind what the vertical model structure localizer is, you can extract from this that all of the Oury inner horn inclusions are local in Ara's model structure (by induction on $C=\Theta_n$). It's somewhat easy to see that you still need the higher dimensional "completeness maps".

Yuki Maehara said he has full results showing that even "bigger" horns than the Oury horns generate the inner anodynes as well, specifically the horns suggested by Joyal and Berger (omitting exactly one codimension 1 subobject), but I don't think he has made his preprint available yet.

A more useful heuristic argument to see why Joyal's proposed model structure doesn't work is that since all strict $\omega$-categories are fibrant in that model structure (they all admit unique inner horn fillers) and the embedding of strict $\omega$-categories is full in cellular sets, it would mean that every pseudofunctor between strict $\omega$-categories is naturally (not pseudonaturally) isomorphic to a strict one, which is trivially false.

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