Theorem 5.3.7 (00A6) · ScalingStacks

Let \(k \geq 0\) and \(n \geq -2\).

The pair (\(n\)-surjective functors, \(n\)-faithful functors) defines a factorization system on the \(\infty\)-category \(\mathrm{Cat}_{(\infty, {k})}\) of \((\infty,k)\)-categories.
This factorization system is compatible with the Cartesian symmetric monoidal structure on \(\mathrm{Cat}_{(\infty, {k})}\).
This factorization system is of small generation. More specifically,
1. if \(n+2 \leq k\) then it is generated by the set \(\{ \partial c_i \rightarrow c_i \}_{n+2 \leq i \leq k}\), and
2. if \(n+2 \geq k\) then it is generated by the single morphism \(\{ \Sigma^k[S^{n-k+1}] \rightarrow\Sigma^k[{\sf pt}] \eqqcolon c_k \}\).

Proof.

In the base case that \(k = 0\), this factorization system is recorded as [Lur09, Ex. 5.2.8.16], which is easy to check is compatible with the Cartesian symmetric monoidal structure and generated by the single morphism \(\{ S^{n+1} \rightarrow{\sf pt}\}\). So, let us assume that \(k > 0\).

If \(n = -2\), then this is the trivial factorization system \((\mathrm{Cat}_{(\infty, {k})},\mathrm{Cat}_{(\infty, {k})}^\simeq)\) (as in Example B.1.7). Moreover, it is trivially compatible with the Cartesian symmetric monoidal structure, and it is also clearly generated by the set \(\{ \partial c_i \rightarrow c_i \}_{0 \leq i \leq k}\): by induction (and the universal property of the categorical suspension functor \(\Sigma[-]\)), a morphism \(\mathcal C\rightarrow\mathcal D\) is right orthogonal to this set if and only if it is an equivalence on maximal subgroupoids and on hom-\((\infty,k-1)\)-categories.

From here, in the case that \(n > -2\) (and \(k > 0\)) the claim follows by applying Theorem B.4.1 inductively (varying both \(k\) and \(n\) simultaneously). ◻