Given a presentably symmetric monoidal \(\infty\)-category \(\mathcal C\) and a small \(\infty\)-operad \(\mathcal O\), the \(\infty\)-category \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is presentable and carries a symmetric monoidal structure. However, it is not necessarily presentably symmetric monoidal: The symmetric monoidal structure on \(\mathrm{Alg}_{\mathcal O}(\mathcal C)\) is not necessarily compatible with finite coproducts (though it is always compatible with sifted colimits). An easy counterexample is \(\mathcal C= \mathrm{Set}\) and \(\mathcal O=\mathbb E_1\).
A.8.10 Presentably \(\mathcal O\)-monoidal \(\infty\)-categories[00J0]
Given an \(\infty\)-operad \(\mathcal O\), a presentably \(\mathcal O\)-monoidal \(\infty\)-category is an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\) such that for every color \(X \in \underline{\mathcal O}\) the \(\infty\)-category \(\mathcal C_X\) is presentable and moreover for every multimorphism \(\{X_i\}_{i \in I} \rightarrow Y\) in \(\mathcal O\) the corresponding multifunctor \(\prod_{i \in I} \mathcal C_{X_i} \rightarrow\mathcal C_Y\) is multi-cocontinuous (i.e. cocontinuous separately in each variable). This is equivalent to the condition that \(\mathcal C\) defines an \(\mathcal O\)-algebra \((\Pr^L,\otimes)\), and we write \(\mathrm{Alg}_\mathcal O(\Pr^L) \subseteq \mathrm{Alg}_\mathcal O(\widehat{\mathrm{Cat}}_\infty)\) for the subcategory whose objects are the presentably \(\mathcal O\)-monoidal \(\infty\)-categories whose morphisms are the \(\mathcal O\)-monoidal left adjoints among them.
Given a presentably \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\in \mathrm{Alg}_\mathcal O(\Pr^L)\), the \(\infty\)-category \(\mathrm{Alg}_\mathcal O(\mathcal C)\) is also presentable. Moreover, the functor \((\mathrm{Op}_{/\mathcal O})^\mathrm{op}\xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}(\mathcal C)} \widehat{\mathrm{Cat}}_\infty\) factors through \(\Pr^R\), i.e. for every morphism \(\mathcal A\rightarrow\mathcal B\) in \(\mathrm{Op}_\mathcal O\) there exists a left adjoint 
the “free \(\mathcal B\)-algebra on an \(\mathcal A\)-algebra” functor [Lur17, Cor. 3.1.3.5]. Of course, these left adjoints then assemble into a functor \(\mathrm{Op}_{/\mathcal O} \xrightarrow{\mathrm{Alg}_{(-)/\mathcal O}} \Pr^L\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2