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](https://mathoverflow.net/questions/29635/what-determines-a-model-structure) 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. 



"Joyal type" model structure for (n,1)-categories?
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.


