Corollary 12.3 (0MJZ) · ScalingStacks

[0MJZ]

Corollary 12.3. A presheaf of sets on $\Delta^{\!\times n}$ is isomorphic to the nerve of a gaunt $n$ -category if and only if it is local with respect to the classes $\mathrm{Segal}_{\Delta^{\!\times n}}$ , $\mathrm{Glob}_{\Delta^{\!\times n}}$ , and $\mathrm{Comp}_{\Delta^{\!\times n}}$ . A presheaf of sets on $\Theta_{n}$ is isomorphic to the nerve of a gaunt $n$ -category if and only if it is local with respect to the classes $\mathrm{Segal}_{\Theta_{n}}$ and $\mathrm{Comp}_{\Theta_{n}}$ .

[0MK0]

Proof. Being local with respect $\mathrm{Segal}_{\Delta^{\!\times n}}$ and $\mathrm{Glob}_{\Delta^{\!\times n}}$ (or to $\mathrm{Segal}_{\Theta_{n}}$ for $\Theta_{n}$ -presheaves) implies that the presheaf is the nerve of a strict $n$ -category. Such an $n$ -category is gaunt if and only if it is local with respect to the morphisms $\sigma^{k}(E)\to\sigma^{k}(C_{0})$ . This last follows from locality with respect to $\mathrm{Comp}_{\Delta^{\!\times n}}$ (or, respectively, with respect to $\mathrm{Comp}_{\Theta_{n}}$ ) because the square

is a pushout square of strict $n$ -categories. ∎