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.

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

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?

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 $Elts_A$.