ScalingStacks

Theorem 3.12. The inclusion $\mathbf{Op}_{d}\hookrightarrow\mathbf{Op}_{\infty}$ admits a left adjoint $h_{d}$, such that for every $\infty$-operad $\mathcal{O}$ the value of $h_{d}$ on $\mathcal{O}$ is the $d$-homotopy operad of $\mathcal{O}$, the unit transformation $\theta_{d}\colon\mathcal{O}\to h_{d}\mathcal{O}$ is essentially surjective, and for all objects $X_{1},\dots,X_{n},Y\in\underline{\mathcal{O}}$, the map of spaces

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

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