Lemma 3.10 (0KF6) · ScalingStacks

[0KF6]

Lemma 3.10. Let $d\geq-1$ and let $\mathcal{O}$ be an $\infty$ -operad. The canonical map $\theta_{d}\colon\mathcal{O}\to h_{d}\mathcal{O}$ is essentially surjective and for all $X_{1},\dots,X_{n},Y\in\underline{\mathcal{O}}$ , the map

\operatorname{Mul}_{\mathcal{O}}\left(\left\{X_{1},\dots,X_{n}\right\};Y\right)\to\operatorname{Mul}_{h_{d}\mathcal{O}}\left(\left\{\theta_{d}\left(X_{1}\right),\dots,\theta_{d}\left(X_{n}\right)\right\};\theta_{d}\left(Y\right)\right)

is a $\left(d-1\right)$ -truncation map.

[0KF7]

Proof. The map $\theta_{d}\colon\mathcal{O}\to h_{d}\mathcal{O}$ is surjective on objects and hence is essentially surjective. For $d\geq 1$ , the second assertion follows from the corresponding fact for $\infty$ -categories; and for $d=-1,0$ , it follows directly from the definition. ∎