ScalingStacks

Definition B.1.12.

Let \(\mathcal O\) be an \(\infty\)-operad and let \(\mathcal C\) be an \(\mathcal O\)-monoidal \(\infty\)-category. Suppose that for every color \(X \in \underline{\mathcal O}\), the \(\infty\)-category \(\mathcal C_X\) of \(X\)-colored objects in \(\mathcal C\) is equipped with a factorization system \((\mathcal L_X,\mathcal R_X)\). We say that the \(\mathcal O\)-monoidal structure of \(\mathcal C\) is compatible with these factorization systems if for every \(n \geq 0\) and every \(n\)-ary operation \((X_1,\ldots,X_n) \rightarrow X\) in \(\mathcal O\), the corresponding functor \(\mathcal C_{X_1} \times \cdots \times \mathcal C_{X_n} \rightarrow\mathcal C_X\) carries morphisms in \(\mathcal L_{X_1} \times \cdots \times \mathcal L_{X_n}\) to morphisms in \(\mathcal L_X\).⁶⁶