ScalingStacks

We first consider the case that \(n + 2 \geq k\), where we need to show that a commuting diagram of \((\infty,k)\)-categories has \((m-n-2)\)-truncated space of lifts. Let \(\alpha\) denote the composite \(\partial c_k = \Sigma^k[\emptyset] \rightarrow\Sigma^k[S^{n-k+1}] \rightarrow\mathcal C\), picking out a pair of parallel \((k-1)\)-morphisms. By observation 5.1.9, the space of lifts of ([00AM]) is equivalent to the space of lift of the following diagram in spaces By definition of faithfulness, if \(G\) is \(m\)-faithful, then the right vertical map is \((m-k)\)-truncated for any \(\alpha \colon \partial c_k \rightarrow\mathcal C\). Hence, it follows from [SY19, Prop. 4.2.8] that the space of lifts of ([00AN]) is \((m-n-2)\)-truncated.