Proposition 4.2.8 (05YN) · ScalingStacks

[05YN]

Proposition 4.2.8. Let $\mathcal{C}$ be a presentable $\infty$ -category. Fix integers $d\geq n\geq-2$ . For every square $q\colon\Delta^{1}\times\Delta^{1}\to\mathcal{C}$ of the form

in which $f\colon A\to B$ is $n$ -connected and $g\colon X\to Y$ is $d$ -truncated, the space of lifts $L\left(q\right)$ is $\left(d-n-2\right)$ -truncated.

[05YP]

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. ∎