2 fixed inaccuracy

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.

1

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 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.