If \(\mathcal C\) is a presentably \(\mathcal O\)-monoidal category for a small operad \(\mathcal O\) and suppose that for every color \(X\in \underline{\mathcal O}\), the presentable \(\infty\)-category \(\mathcal C_{X}\) is equipped with a factorization system \((\mathcal L_X, \mathcal R_X)\) generated by a set \(S_X\). Then, the factorization systems are compatible with the \(\mathcal O\)-monoidal structure if and only if for every operation \((X_1, \ldots, X_n) \rightarrow X\) in \(\mathcal O\), the corresponding functor \(\mathcal C_{X_1} \times \cdots \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) carries morphisms in \(S_{X_1} \cdots \times \cdots S_{X_n}\) to morphisms in \(\mathcal L_X\).
Proof.
Assume that \(\mathcal C_{X_1} \times \cdots \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) carries morphisms in \(S_{X_1} \cdots \times \cdots S_{X_n}\) to morphisms in \(\mathcal L_X\). By assumption, the functor \(\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) preserves small colimits separately in all variables. Since for every \(Y \in \underline{\mathcal O}\), the class of morphisms \(\mathcal L_Y\) is by proposition B.1.14 the smallest saturated class of morphisms in \(\mathcal C_Y\) that contains \(S_Y\), the functor \(\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_{X}\) therefore also carries morphisms in \(\mathcal L_{X_1} \times \cdots \times \mathcal L_{X_n}\) to morphisms in \(\mathcal L_X\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2