Lemma 4.3.2 (05YQ) · ScalingStacks

[05YQ]

Lemma 4.3.2. Let $n\geq-2$ and let $\mathcal{C}$ be a presentable $\infty$ -category. If a morphism $f\colon A\to B$ is $n$ -connected, then it is $\left(n-\frac{1}{2}\right)$ -connected.

[05YR]

Proof. By the Yoneda lemma it is enough to show that for every $n$ -truncated object $Z$ in $\mathcal{C}$ the induced map

f_{*}\colon\operatorname{Map}\left(B,Z\right)\to\operatorname{Map}\left(A,Z\right)

is an equivalence. For this, it is enough to show that for every $g\colon A\to Z$ , the fiber of $f_{*}$ over $g$ is contractible. By T.5.5.5.12, the fiber is equivalent to the space of lifts for the square

which is contractible by definition as $f\colon A\to B$ was assumed to be $n$ -connected. ∎