ScalingStacks

Proof. The 0-truncated objects of $\pre(\Upsilon_{n})$ are precisely those presheaves of spaces taking values in the 0-truncated spaces, i.e., functors $\Upsilon_{n}^{\mathrm{op}}\to\set$. The 0-truncated objects of $\cat_{(\infty,n)}=S^{-1}\pre(\Upsilon_{n})$ consist of precisely those 0-truncated objects of $\pre(\Upsilon_{n})$ which are $S$-local. By Lemma 6.6, the nerve of every gaunt $n$-category is $S$-local, and so the result follows. ∎