ScalingStacks

Proof. The nerve of a gaunt $n$-category is $S$-local (cf. Lemma 6.6). Conversely, for any $X\in\tau_{\leq 0}\cat_{(\infty,n)}\subseteq\Fun(\Upsilon_{n}^{\mathrm{op}},\set)$, we may restrict to $\mathbb{G}_{n}$ to obtain a globular set $H_{X}$. For $0\leq j<i\leq n$, apply $X$ to the unique nondegenerate $i$-cell $\mu\colon C_{i}\to C_{i}\cup^{C_{j}}C_{i}$ connecting the initial and terminal vertices; this gives rise to the various compositions

$$X(C_{i})\times_{X(C_{j})}X(C_{i})\cong X(C_{i}\cup^{C_{j}}C_{i})\to X(C_{i}).$$

By examining the maps

$$X(C_{i})\times_{X(C_{j})}X(C_{i})\times_{X(C_{j})}X(C_{i})\cong X(C_{i}\cup^{C_{j}}C_{i}\cup^{C_{j}}C_{i})\to X(C_{i})$$

corresponding to the unique nondegenerate $i$-cell

$$C_{i}\to C_{i}\cup^{C_{j}}C_{i}\cup^{C_{j}}C_{i}$$

connecting the initial and terminal vertices, we find that these compositions are associative, and by examining the maps $X(C_{j})\to X(C_{i})$ induced by the nondegenerate cell $C_{i}\to C_{j}$, we find that these compositions are unital. From this we deduce that $H_{X}$ forms a strict $n$-category. Finally, since $X$ is local with respect to $K_{k}\to C_{k}$, it follows that $H_{X}$ is gaunt. Now map $A\to X$, with $A\in\Upsilon_{n}$ induces a map $A\to\nu H_{X}$, and hence we have a map $X\to\nu H_{X}$ in $\tau_{\leq 0}\cat_{(\infty,n)}$. By construction this is a cellular equivalence, whence $X\simeq\nu H_{X}$. ∎