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:

  1. Is there a good reason why one shouldn't use this model structure?
  2. Is this model structure Quillen equivalent to the usual model structure on $\mathsf{RelCat}$, constructed by Barwick and Kan, i.e. to $\infty$-categories?
  3. 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.
  • $\begingroup$ If it's a left induced model structure, shouldn't it be the case that (acyclic) cofibrations are simply maps that become (acyclic) cofibrations in Cat when applying h? I think fibrations are harder. In general, not knowing if it's cofibrantly generated would be a reason not to use it. To answer (2), see if there is a containment of cofibrations. If so, the identity is a Quillen equivalence. $\endgroup$ – David White Aug 16 '16 at 0:55
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    $\begingroup$ It doesn't have the right weak equivalences! An equivalence of $(\infty,1)$-categories is not just a functor inducing an equivalence on homotopy categories. For example, the functor $C \to ihC$ is not an equivalence if $C$ is not in the image of $i$. $\endgroup$ – AAK Aug 16 '16 at 1:56
  • $\begingroup$ In answer to 1., whether or not to use a particular model category structure really depends on what you want to use it for! It is perhaps more useful to think of the model category structure as a tool to do other things rather than an end in itself. That second approach is the next level, i.e. studying the tools! $\endgroup$ – Tim Porter Aug 16 '16 at 7:14
  • $\begingroup$ @AdeelKhan Aha! This is my fundamental misunderstanding! I wonder if it's possible to tweak things to work. Maybe by restricting morphisms to be adjunctions or something. $\endgroup$ – Tim Campion Aug 16 '16 at 11:51

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