ScalingStacks

Lemma 2.4.2. Let $\mathcal{P}$ be a reduced $\infty$-operad and let $\mathcal{C}$ be a presentably symmetric monoidal $\infty$-category. The forgetful functor

$$U_{\mathcal{P}}\colon\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathbf{Triv}}\left(\mathcal{C}\right)\simeq\mathcal{C}$$

admits a left adjoint $F_{\mathcal{P}}$ and the associated monad $T_{\mathcal{P}}=U_{\mathcal{P}}\circ F_{\mathcal{P}}$ acts on an object $X\in\mathcal{C}$ as follows:

$$T_{\mathcal{P}}\left(X\right)=U_{\mathcal{P}}F_{\mathcal{P}}\left(X\right)=\operatorname*{colim}\mathcal{P}_{\boldsymbol{\mathbf{SSeq}}}\left(X\right)=\coprod_{n\geq 0}\left(\mathcal{P}\left(n\right)\otimes X^{\otimes n}\right)_{h\Sigma_{n}}$$

(where we let $\otimes$ denote the canonical enrichment of $\underline{\mathcal{C}}$ over $\mathcal{S}$ as well).