ScalingStacks

Theorem B.4.1.

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

1. The \(\infty\)-category \(\mathrm{Cat}[\mathbb V]\) of \(\mathbb V\)-enriched \(\infty\)-categories admits a factorization system \((\mathcal L_\mathrm{Cat},\mathcal R_\mathrm{Cat})\), described as follows.

   1. A morphism lies in \(\mathcal L_\mathrm{Cat}\) if and only if it is surjective on objects (i.e. \(\iota_0\)-surjective) and lies in \(\mathcal L\) homwise.

   2. A morphism lies in \(\mathcal R_\mathrm{Cat}\) if and only if it lies in \(\mathcal R\) homwise.

2. If \(S\) is a set of generators for \(\mathcal L\), then the localization of \(\Sigma[S]\) is a set of generators for \(\mathcal L_\mathrm{Cat}\).

3. If \(\mathbb V\) is symmetric monoidal, then this factorization system is compatible with the resulting symmetric monoidal structure on \(\mathrm{Cat}[\mathbb V]\).