[00KG]
Lemma B.2.2.
Let \(\mathcal C\) be an \(\infty\)-category equipped with a factorization system \((\mathcal L,\mathcal R)\), let \(\mathcal C_0 \subseteq \mathcal C\) be a subcategory, and suppose that \(\mathcal L_0 \coloneqq \mathcal L\cap \mathcal C_0\) and \(\mathcal R_0 \coloneqq \mathcal R\cap \mathcal C_0\) fulfill conditions (1) and (2) of observation B.2.1, so that they induce a factorization system \((\mathcal L_0, \mathcal R_0)\) on \(\mathcal C_0\). Then we have:
If \(\mathcal C_0\) and \(\mathcal C\) are presentable, \((\mathcal L, \mathcal R)\) is of small generation, and the inclusion \(\mathcal C_0 \rightarrow\mathcal C\) admits a left adjoint \(L\colon \mathcal C\rightarrow\mathcal C_0\), then \((\mathcal L_0, \mathcal R_0)\) is of small generation and \(L(\mathcal L) \subseteq \mathcal L_0\).
If \(\mathcal C_0\) and \(\mathcal C\) are furthermore presentably \(\mathcal O\)-monoidal for a small operad \(\mathcal O\), the left adjoint \(L\colon \mathcal C\rightarrow\mathcal C_0\) is \(\mathcal O\)-monoidal, and \((\mathcal L, \mathcal R)\) is of small generation and compatible with the \(\mathcal O\)-monoidal structure, then so is \((\mathcal L_0, \mathcal R_0)\).
[00KH]
Proof.
For (1), let \(L\colon \mathcal C\rightarrow\mathcal C_0\) denote the left adjoint, let \(S\) be a generating set for \(\mathcal L\) and define \(S_0\) to be the set of \(\mathcal C_0\)-morphisms \(S_0 \coloneqq L(S)\). By adjunction, a morphism \(f\) in \(\mathcal C_0\) is in \(S_0^{\bot}\) iff it is in \(S^{\bot} = \mathcal R\), and hence that \(S_0^{\bot} = \mathcal R\cap \mathcal C_0,\) proving that \((\mathcal L_0, \mathcal R_0)\) is generated by \(S_0\). \(L(\mathcal L) \subseteq \mathcal L_0\) follows from the adjunction.
For (2), it follows from the proof of (1) that for every \(X\in \underline{\mathcal{O}}\), the class \((\mathcal L_0)_X\) is generated by \((S_0)_X \coloneqq L_X(S_X)\) where \(S_X\) is a generating set for \(\mathcal L_X\). Hence, to check that \((\mathcal L_0, \mathcal R_0)\) is compatible with the \(\mathcal O\)-monoidal structure, it suffices by lemma B.1.18 to show that for every operation \((X_1, \ldots, X_n) \rightarrow X\) in \(\mathcal O\), the induced functor \((\mathcal C_0)_{X_1}\times \cdots \times (\mathcal C_0)_{X_n} \rightarrow(\mathcal C_0)_{X}\) carries morphisms in \(L_{X_1}(S_{X_1}) \times \cdots \times L_{X_n}(S_{X_n})\) to a morphism in \((\mathcal L_0)_X\). But since \(L\) is \(\mathcal O\)-monoidal, such a family of morphism is carried to the image under \(L_X\) of their product in \(\mathcal C_X\). Since \(\mathcal L\) is compatible with the monoidal structure, that product is in \(\mathcal L_X\) and hence the morphisms are carried to a morphism in \(L_X(\mathcal L_X) \subseteq (\mathcal L_0)_X\). ◻