ScalingStacks

A.8.6 Symmetric monoidal structure on overcategories[00IV]

Let \(\mathcal C\) be a symmetric monoidal \(\infty\)-category and \(A \in \mathrm{CAlg}(\mathcal C)\) be a commutative algebra object therein. Then, there exists a symmetric monoidal structure on the overcategory \(\mathcal C_{/A}\) ([Lur17, Thm. 2.2.2.4]), universally characterized (cf. [Lur17, Def. 2.2.2.1]) by the following equivalence of \(\infty\)-categories for any \(\infty\)-operad \(\mathcal O\) \[\underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C_{/A}) \simeq \underline{\mathrm{Alg}}_{\mathcal O}(\mathcal C)_{/A},\] where on the right hand side we \(A\) is equipped with the \(\mathcal O\)-algebra structure induced by the terminal map of operads \(\mathcal O^{\otimes} \rightarrow\mathrm{Comm}\).