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.)

Answer 2

The fact that G is a test category falls in large class of examples. Grothendieck proved that a small category $A$ is a local test category if and only if there exists a presheaf $I$ on $A$ which is an interval (i.e. which has two disjoint global sections) such that, for any representable presheaf $a$, the cartesian product $a\times I$ is aspherical (a presheaf $X$ is aspherical if the classifying space of its category of elements is contractible; for instance, any representable presheaf is aspherical because any category with a terminal object is contractible). A small category $A$ is a test category if and only if it is a local test category with contractible classifying space. For instance, any small category with a terminal object, with finite products, and with a representable interval is a test category. This is the case for the category of non-empty finite sets; see Theorem 1.5.6 and Corollary 1.5.7 in Maltsiniotis book.

I just would like to point out two nice things about Reid's last remark: the category of $1$-groupoids is canonically equivalent to the full subcategory of the category of presheaves on $G$ spanned by the presheaves which satisfy the (strict) Segal condition. This imply that the "classical model structure" and the "Joyal model structure" coincide on $\widehat{G}$, and define the homotopy theory of $(\infty,0)$-categories (aka $\infty$-groupoids). If we consider the Joyal model structure on $\widehat{\Delta}$, the Quillen adjunction with $\widehat{G}$ then really extends the adjunction between categories and groupoids.

I cannot resist to assert that such a picture can be pushed higher: we can replace $\Delta$ by Joyal's categories $\Theta_n$ (or even by some category of operators $\Theta_A$, corresponding to a contractible $n$-operad à la Batanin), and then produce an analog of $G$ with respect to $\Theta_n$, in such a way that we shall get the same picture, but relating $(\infty,n)$-categories to $(\infty,0)$-categories. The construction of such symmetrization of (weakenings) of Joyal's categories and the proof that they lead to test categories is the subject of the PhD thesis (in preparation) of Dimitri Ara (Paris 7). Such symmetrizations have been constructed explicitely by Grothendieck at the very begining of Pursuing stacks, and lead to a definition of weak $\infty$-groupoids, very close to Batanin's notion of weak higher categories; see these two preprints (1 2) of Maltsiniotis (in French). Grothendieck's conjecture that weak $\infty$-groupoids model homotopy types is stated very explicitely using this very definition of higher groupoids. I guess this is one of the starting points/motivations of his theory of test categories.