[Math] Are non-empty finite sets a Grothendieck test category

Question

A "test category" is a certain kind of small category $A$ which turns out to have the following property: the category $\widehat{A}$ of presheaves of sets on $A$ admits a model category structure, which is Quillen equivalent to the usual model category structure on spaces.

The notion of test category was proposed by Grothendieck, and the above result was proved by Cisinski (Les préfaisceaux comme modèles des types d'homotopie. Astérisque No. 308 (2006)).
Examples of test categories include the category $\Delta$ of non-empty finite ordered sets (i.e., the indexing category for simplicial sets), and $\square$, the indexing category for cubical sets.

It's hard for me to give the precise definition of test category here: it involves the counit of an adjunction $i_A: \widehat{A} \rightleftarrows \mathrm{Cat} :i_A^*$, where the left adjoint $i_A$ sends a presheaf $X$ to the comma category $A/X$ (where we think of $A\subset \widehat{A}$ by yoneda). An online introduction to test categories, which includes the full definition and an account of Cisinski's results, is given in Jardine, "Categorical homotopy theory".

I don't really understand how one should try to prove that a particular category is a test category. The example I have in mind is $G$, the (skeleton of) the category of non-empty finite sets, and all maps between them. I believe this should be a test category; is this true?

Note that there is a "forgetful functor" $\Delta\rightarrow G$, which induces some pairs of adjoint functors between $\widehat{\Delta}$ and $\widehat{G}$. If $G$ is really a test category, I would expect one of these adjoint pairs to be a Quillen equivalence.

Another note: $G$ is equivalent to the category of finite, contractible groupoids, which is how I am thinking about it.

Answer 1

That your G is a test category is stated in the last sentence of 4.1.20 in the paper of Cisinski you mention. This case is also treated in more detail in section 8.3, where it is shown that the left Kan extension/restriction along both adjunctions induced by your "forgetful functor" are Quillen equivalences (8.3.8).

(By the way, if I recall correctly, it is a corollary that if we give simplicial sets the Joyal model structure instead, then both adjunctions are still Quillen pairs, and they realize the two adjoints to the inclusion of (∞,0)-categories in (∞,1)-categories.)