ScalingStacks

Proof. We prove this by induction on $d$. For $d=n$, the claim follows from the definition of an $n$-connected morphism and the fact that a space is $\left(-2\right)$-connected if and only if it is contractible. We now assume that this is true for $d-1$, and prove it for $d$. Denote the space of lifts by $L\left(q\right)$. By T.5.5.6.15, it suffices to show that the diagonal map $\delta\colon L\left(q\right)\to L\left(q\right)\times L\left(q\right)$ is $\left(d-n-3\right)$-truncated. By 4.1.5, the homotopy fiber over a point $\left(s_{0},s_{1}\right)\in L\left(q\right)\times L\left(q\right)$ is equivalent to the space of lifts in the square

where the bottom map is $\left(s_{0},s_{1}\right)$. By T.5.5.6.15, since $X\to Y$ is $d$-truncated, $X\to X\times_{Y}X$ is $\left(d-1\right)$-truncated and, therefore, by induction, the space of lifts is $\left(\left(d-1\right)-n-2\right)$-truncated and we are done. ∎