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The Joyal model structure on the category of simplicial sets, has monomorphisms as cofibrations and quasi-categories as fibrant objects (these model $(\infty,1)$-categories). In HTT (section 2.3.4) Lurie defines the notion of an $n$-category (these model $(n,1)$-categories) and proves some theorems about them. For example, $1$-categories are just simplicial sets which are isomorphic to a nerve of a category (prop. 2.3.4.5). Moreover, every quasi-category $\mathcal{C}$ has an $n$-truncation $h_n\mathcal{C}$ which is an $n$-category with a map $\mathcal{C} \to h_n\mathcal{C}$ which is universal from $\mathcal{C}$ to an $n$-category (see prop. 2.3.4.12 for a precise statement).

Qusetion: Is there a model structure on the category of simplicial sets with monomorphisms as cofibrations, $n$-categories as fibrant objects and such that two quasi-categories are equivalent in this model structure if and only if their $n$-truncations are equivalent in the Joyal model structure?

I ask this mainly out of pure curiosity and haven't put too much effort in thinking about it myself, in the hope that someone already knows the answer. Hence, perhaps there are simple reasons that this can't happen, in this case (and any other case), I'm open to considering some variations.

Remark: In fact, as I learned from this question, the cofibrations and fibrant objects determine the model strucute, so the question can be broken in two. First, whether there is a model structure with the specified cofibrations and fibrant objects, and if so, how can we interpret the weak equivalences.

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I don't know if this will work with the definition of $(n, 1)$-categories exactly as stated by Lurie, but it will work with a reasonable modification.

Definition 2.3.4.1 from HTT basically says that a quasicategory is an $(n, 1)$-category if it has no non-trivial morphisms above dimension $n$ and it insists that this condition is satisfied on the nose. This is probably too much to ask if we aim for a nice model for the homotopy theory of $(n, 1)$-categories. (In particular, the definition is not invariant under categorical equivalences.) We would rather impose this condition homotopically.

It is perfectly sensible to define an $(n, 1)$-category as a quasicategory whose mapping spaces are $(n-1)$-truncated. This can be concisely stated as follows. An $(n, 1)$-category is a quasicategory $\mathcal{C}$ with the right lifting property with respect to inclusions $\partial\Delta[m] \to \Delta[m]$ for all $m \ge n + 2$. I'm pretty sure that this is equivalent to $\mathcal{C}$ being categorically equivalent to an $(n, 1)$-category in the sense of HTT.

Now we can simply take the left Bousfield localization of the Joyal model structure with respect to the maps $\partial\Delta[m] \to \Delta[m]$ for $m \ge n + 2$. This exists by the general theory of Bousfield localizations (see e.g. Hirschhorn's book) and has $(n, 1)$-categories as fibrant objects, monomorphisms as cofibrations and weak equivalences between fibrant objects are exactly the categorical equivalences.

I'm not aware of any reference that constructs this model category, much less one that develops it any further.

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    $\begingroup$ I have just remembered that Rezk considers the Segal space version of this problem in A Cartesian Presentation of Weak n-categories. In fact, he covers general $(n, k)$-categories not just $(n, 1)$-categories. He uses a similar Bousfield localization of the model structure for complete $\Theta_k$-spaces. $\endgroup$
    – Karol Szumiło
    Commented Jan 20, 2017 at 19:52
  • $\begingroup$ It is true that an $\infty$-category is equivalent to an $n$-category iff its mapping spaces are $(n-1)$-truncated (homotopically). I don't see why (and I don't think it's true that) it is equivalent to the lifting property you suggest. Being equivalent to a 1-category is not just being 2-co-skeletal, right? $\endgroup$
    – KotelKanim
    Commented Jan 21, 2017 at 8:17
  • $\begingroup$ Anyway, I suppose there is a way to rephrase it using some lifting property, which you can then plug into the Bousfield localization machine. I did hope for a strict version though. As you say, its weak equivalences would be incomparable with categorical equivalences, hence it will not be a localization of the Joyal model structure. $\endgroup$
    – KotelKanim
    Commented Jan 21, 2017 at 8:24
  • $\begingroup$ Why do you believe that the lifting property I gave is not equivalent to $(n - 1)$-truncated mapping spaces? It is fairly easy to see, e.g., you can assume that your quasicategory is the coherent nerve of a fibrant simplicial category and use the left adjoint $\mathfrak{C}$. $\endgroup$
    – Karol Szumiło
    Commented Jan 21, 2017 at 23:01
  • $\begingroup$ The strict definition of $(n, 1)$-categories is indeed not invariant under categorical equivalences. If you want to use it to construct a Bousfield localization from it, what model structure would you localize? I don't see any candidate other than the Joyal model structure and if you try that with the Joyal model structure you will end up with the same localization that I described anyway. $\endgroup$
    – Karol Szumiło
    Commented Jan 21, 2017 at 23:04
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You may be interested in having a look at this: https://arxiv.org/abs/1810.11188.

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