ScalingStacks

A.8.5 \(\mathcal O\)-algebras of symmetric monoidal \(\infty\)-categories[00IU]

Let \(\mathcal O, \mathcal C\) be \(\infty\)-operads, then the \(\infty\)-category \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) has the structure of an \(\infty\)-operad [Lur17, Ex. 3.2.4.4]. From now on, we will denote the \(\infty\)-operad of \(\mathcal O\)-algebras in \(\mathcal C\) by \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\), and the underlying \(\infty\)-category of \(\mathcal O\)-algebra by \(\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C)\), or just \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) if clear from context. When the \(\infty\)-operad \(\mathcal C\) is in fact a symmetric monoidal \(\infty\)-category, i.e. \(\mathcal C^{\otimes} \rightarrow\mathrm{Fin}_*\) is a coCartesian fibration, then so is the \(\infty\)-operad \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\). Furthermore, let \(X \in \underline{\mathcal O}\) be a color, the evaluation functor \(e_X \colon \underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C) \rightarrow\underline{\mathcal C}\), which takes an \(\mathcal O\)-algebra to its underlying \(X\)-object, is symmeric monoidal [Lur17, Prop. 3.2.4.3]. More generally, for any map of \(\infty\)-operads \(\mathcal O' \rightarrow\mathcal O\), the pullback functor on algebras \(\mathrm{Alg}_{\mathcal O}(\mathcal C) \rightarrow\mathrm{Alg}_{\mathcal O'}(\mathcal C)\) is a symmetric monoidal functor.