One can left-induce a model structure from $\mathsf{Cat}$ to $\mathsf{RelCat}$ along the "homotopy category" functor. The weak equivalences are the relative functors which induce equivalences of homotopy categories, which is good. The cofibrations are the injective-on-objects relative functors, and the acyclic fibrations are the equivalences which are retracts in $\mathsf{RelCat}$. If I'm not mistaken, the existence of this model structure follows from the main result of Hess, Kedziorek, Riehl, and Shipley.

(In case it's not clear, $\mathsf{Cat}$ is the category of categories, $\mathsf{RelCat}$ is the category of relative categories (the "simplest" model of $\infty$-categories), and we're left-inducing the canonical model structure on $\mathsf{Cat}$ along the adjunction $h: \mathsf{RelCat} \overset{\to}{\leftarrow} \mathsf{Cat}: i$, where $iC$ is the relative category with underlying category $C$ with just the isomorphisms being weak equivalences and $hD$ is the homotopy category of the relative category $D$.)

In some ways, this model structure is charmingly simple, and it has the right weak equivalences, but I can't find mention of it in Barwick and Kan. I also don't know a good generating set of acyclic cofibrations, nor do I really understand fibrations. So a few questions:

- Is there a good reason why one shouldn't use this model structure?
- Is this model structure Quillen equivalent to the usual model structure on $\mathsf{RelCat}$, constructed by Barwick and Kan, i.e. to $\infty$-categories?
- Is there a good description of the acyclic cofibrations / fibrations in this model structure? For example, I would guess that an object is fibrant iff its weak equivalences are saturated in the sense that they are precisely the maps inverted upon passage to the homotopy category.