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

By examining the maps

corresponding to the unique nondegenerate $i$-cell

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 $X\simeq\nu H_{X}$. ∎

Proof. We observe that the existence of an equivalence $\tau_{\leq 0}\mathcal{C}\simeq\gaunt_{n}$, Lemma 4.5, Lemma 4.10, and Corollary 4.14 guarantee that $\Aut((\tau_{\leq 0}\mathcal{C})^{\omega})$ in $\Fun((\tau_{\leq 0}\mathcal{C})^{\omega},(\tau_{\leq 0}\mathcal{C})^{\omega})$ is equivalent to the discrete group $(\mathbb{Z}/2)^{\times n}$. It therefore suffices to exhibit an equivalence of $\infty$-categories $\Aut(\mathcal{C})\simeq\Aut((\tau_{\leq 0}\mathcal{C})^{\omega})$.

Clearly $\Aut(\mathcal{C})$ is contained in the full subcategory $\Fun^{\mathrm{L}}(\mathcal{C},\mathcal{C})\subset\Fun(\mathcal{C},\mathcal{C})$ spanned by those functors that preserve small colimits. Since $(\tau_{\leq 0}C)^{\omega}$ is dense in $\mathcal{C}$, it follows from Lemma 9.2 that the inclusion $(\tau_{\leq 0}C)^{\omega}\hookrightarrow C$ induces a fully faithful functor

Moreover, any autoequivalence of $\mathcal{C}$ restricts to an autoequivalence of $\tau_{\leq 0}\mathcal{C}$ and hence an autoequivalence of $(\tau_{\leq 0}\mathcal{C})^{\omega}$. Thus restriction furnishes us with a fully faithful functor from $\Aut(\mathcal{C})$ to $\Aut((\tau_{\leq 0}\mathcal{C})^{\omega})\simeq(\mathbb{Z}/2)^{n}$.

It remains to show that the restriction functor is essentially surjective. For this, suppose $(\tau_{\leq 0}\mathcal{C})^{\omega}\to(\tau_{\leq 0}\mathcal{C})^{\omega}$ an autoequivalence. One may form the left Kan extension $\Phi\colon\mathcal{C}\to\mathcal{C}$ of the composite

along the inclusion $(\tau_{\leq 0}\mathcal{C})^{\omega}\hookrightarrow\mathcal{C}$. One sees immediately that $\Phi$ is an equivalence, and moreover its restriction to $(\tau_{\leq 0}\mathcal{C})^{\omega}$ coincides with $\phi$. ∎