Lemma B.4.3 (00LH) · ScalingStacks

Fix a presentably monoidal \(\infty\)-category \(\mathbb V\) equipped with a compatible factorization system \((\mathcal L,\mathcal R)\).

The \(\infty\)-category \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\) of categorical \(\mathbb V\)-algebras admits a factorization system \((\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}},\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}})\), described as follows.
1. A morphism lies in \(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}}\) if and only if it is an \(\iota_0\)-equivalence and it lies in \(\mathcal L\) homwise.
2. A morphism lies in \(\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}\) if and only if it lies in \(\mathcal R\) homwise.
If \(S\) is a set of generators for \(\mathcal L\), then \(\Sigma[S] \coloneqq \{ \Sigma(s)\}_{s \in S}\) is a set of generators for \(\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}}\).
If \(\mathbb V\) is symmetric monoidal, then the factorization system \((\mathcal L_{\mathrm{Alg}_{\mathrm{Cat}}},\mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}})\) is compatible with the resulting symmetric monoidal structure on \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V]\).

[00LL]

Proof.

By definition, the Cartesian fibration \(\mathrm{Alg}_{\mathrm{Cat}}[\mathbb V] \xrightarrow{\iota_0} \mathcal S\) is the unstraightening of a composite functor \[\mathcal S^{\mathrm{op}} \xrightarrow{{\textup{codisc}}} (\mathrm{Op}_{/\mathbb E_1})^{\mathrm{op}} \xrightarrow{\mathrm{Alg}_{(-)/\mathbb E_1}(\mathbb V)} {\Pr}^R ~.\]^* For a space \(X \in \mathcal S\), its corresponding \(\infty\)-operad \({\textup{codisc}}(X) \in \mathrm{Op}_{/\mathbb E_1}\) has space of colors given by pairs of points \(x,y \in X\) (up to a symmetrization (i.e. the quotient by the \(\mathfrak S_2\)-action) coming from [Lur17, Thm. 4.1.3.14]), and a categorical \(\mathbb V\)-algebra \(\mathcal C\) with space of objects \(X\) assigns to these the hom-object \(\mathrm{Hom}_\mathcal C(x,y) \in \mathbb V\). Hence, checking conditions on morphisms in \(\underline{\mathbb V}\) colorwise over \({\textup{codisc}}(X)\) indeed corresponds to checking conditions on morphisms homwise, and thereafter part ([00LI]) follows by combining Theorem B.3.1.([00L7]) and Lemma B.2.4.

Thereafter, part ([00LJ]) follows from the observation that \((\Sigma[S])^\bot = \mathcal R_{\mathrm{Alg}_{\mathrm{Cat}}}\), which is immediate from the universal property of \(\Sigma[-]\).

Lastly, part ([00LK]) follows from the assumption that \((\mathcal L,\mathcal R)\) is compatible with the monoidal structure of \(\mathbb V\). ◻