Is the Thomason model structure the optimal realization of Grothendieck's vision?

Question

In Pursuing Stacks, Grothendieck uses the category $Cat$ of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy theory, i.e. with weak homotopy equivalences $\mathcal W_{min}$ -- those functors which become weak homotopy equivalences upon taking classifying spaces.

The Thomason model structure is indeed a model structure on $Cat$ with exactly these weak equivalences. But it's a little strange. The natural idea is to transfer the model structure from $sSet$ along the nerve / realization adjunction $c \dashv N$, but this doesn't quite work so instead one transfers along $c \, sd^2 \dashv Ex^2 N$ where $sd \dashv Ex$ is the barycentric subdivision adjunction.

But Grothendieck envisioned a much more systematic connection between presheaf categories and $Cat$. For a small category $A$, he viewed the "category of elements" functor $Elts_A : Psh(A) \to Cat$ and its right adjoint as the fundamental way to endow $Psh(A)$ with a notion of weak equivalence. Indeed, he showed (see Cisinski's Les Prefaisceaux comme modeles des types d'homotopie Section 4.2) that if $A$ is a test category, then there is a model structure on $Psh(A)$ with cofibrations the monomorphisms and weak equivalences given by $Elts_A^{-1}(\mathcal W_{min})$, which models the homotopy theory of spaces. For example, the Kan-Quillen model structure on $sSet$ is of this form.

Questions: Let $A$ be a test category.

1. Is there a model structure on $Cat$ induced projectively along $Elts_A$ from the Grothendieck model structure on $Psh(A)$?

2. Is there a model structure on $Cat$ such that the Grothendieck model structure on $Psh(A)$ is induced injectively along $Elts_A$ from this model structure?

3. If the answer to both (1) and (2) is "yes", then is this perhaps one and the same model structure, possibly even independent of $A$?

Note that $Elts_A$ is the "canonical test functor". The categorical realization functor $c: sSet \to Cat$ is another test functor, but needs to be "improved" to $c sd^2$ before a model structure can be transferred. So one might also ask for which test functors such a model structure can be transferred, and apparently the answer is "not all of them". The hope is that things work out in the canonical case of $Elts_A$.

Answer 1

The answer to question 1) is no. However, the functor $$N Elts_A:Psh(A)\to sSet$$ commutes with colimits and is a left Quillen equivalence whenever $A$ is a test category. We may transfer the model structure on $Psh(A)$ to $Cat$ through the left adjoint $$c Sd N Elts_A:Psh(A)\to Cat$$ (the proof is slight variation on Thomason's original argument). Such a model structure will be quite similar to Thomason's original construction (in particular, all cofibrations will be retracts of Dwyer maps), but it will definitely depend on $A$ (for any weakly contractible category $C$, the product $A\times C$ is again a test category, and playing with this freedom on $C$, you can produce cofibrations which cannot be reached by Thomason's original definition).

Here is another variation. If $A$ is morevoer an elegant Reedy category, you may do the same using the functor $$c Sd Sd' : Psh(A)\to Cat$$ where the subdivision functor $$Sd':Psh(A)\to sSet$$ is the left Kan extension of the functor $A\to sSet$ which sends an object $a$ to the nerve of the partially ordered set of non-empty subobjects of the presheaf represented by $a$. For $A=\Delta$, you will recover exactly Thomason's model structure, but you may play around with variants of cubical sets, Joyal's $\Theta$ and so on.

The answer to question 2) is unknown to me. Note however that, for any (local) test category $A$, the Grothendieck model structure is indeed transfered injectively through the functor $$N Elts_A:Psh(A)\to sSet$$ from the classical model structure on simplicial sets.

Finally, Grothendieck was interested by the existence of a model structure on $Cat$ only for the purpose of proving the existence of homotopy limits intrinsically in $Cat$. He knew how to construct homotopy pullback by other means, using his theory of smooth and proper functors though. For homotopy colimits, he always used the description via lax colimits à la Thomason, which is indeed much more efficient than what we obtain by applying the general machinery to the Thomason model category. Furthermore, Grothendieck is very explicit in Pursuing stacks about the fact that in a model structure, only the class of weak equivalences matters, and that we may always modify cofibrations according to our needs. For instance, one may modify Thomason's model structure so that the functor $$C\mapsto C^{op}$$ is a left Quillen equivalence. We may also modify it further in order to force all fibrations to be both smooth and proper.