Lemma 5.1.2 (05Z9) · ScalingStacks

[05Z9]

Lemma 5.1.2. Let $\mathcal{C}$ and $\mathcal{D}$ be symmetric monoidal $\infty$ -categories and let $F\colon\mathcal{C}\to\mathcal{D}$ be a symmetric monoidal functor. If $\underline{F}\colon\underline{\mathcal{C}}\to\underline{\mathcal{D}}$ is a left adjoint, then the induced functor $F^{\otimes}\colon\mathcal{C}^{\otimes}\to\mathcal{D}^{\otimes}$ is a left adjoint relative to $\mathbf{Fin}_{*}$ and for every $\infty$ -operad $\mathcal{P}$ the induced functor $\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)\to\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{D}\right)$ is a left adjoint.

[05ZA]

Proof. For every $\left\langle n\right\rangle\in\mathbf{Fin}_{*}$ , the restriction of $F^{\otimes}$ to the fiber over $\left\langle n\right\rangle$ is just $F^{n}\colon\mathcal{C}^{n}\to\mathcal{D}^{n}$ , which is clearly a left adjoint. Hence, by A.7.3.2.7, the functor $F^{\otimes}$ is a left adjoint relative to $\mathbf{Fin}_{*}$ . Let $G^{\otimes}$ be the right adjoint of $F^{\otimes}.$ Applying A.7.3.2.13, we obtain that $F^{\otimes}$ and $G^{\otimes}$ induce an adjunction:

\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{C}\right)\leftrightarrows\underline{\operatorname{Alg}}_{\mathcal{P}}\left(\mathcal{D}\right).

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