ScalingStacks

Corollary 5.3.13.

Given \(j \geq k \geq 0\) and either \(n \geq j\) or \(k>n \geq -2\), and \(F \colon \mathcal A\rightarrow\mathcal B\) in \(\mathrm{Cat}_{(\infty, {j})}\). Then, the maximal sub-\((\infty,k)\)-category functor \(\iota_k \colon \mathrm{Cat}_{(\infty, {j})} \xrightarrow{\iota_k} \mathrm{Cat}_{(\infty, {k})}\) preserves \(n\)-factorizations. That is, if \(\mathrm{Fact}_n (F)\) is the factorization of \(F\) with respect to the (\(n\)-surjective, \(n\)-faithful) factorization system, then the induced factorization \(\iota_k \mathcal A\rightarrow\iota_k \mathrm{Fact}_n(F) \rightarrow\iota_k \mathcal B\) realizes \(\iota_k \mathrm{Fact}_n(F)\) as the factorization \(\mathrm{Fact}_n (\iota_k F)\) of \(\iota_k F\) with respect to the (\(n\)-surjective, \(n\)-faithful) factorization system on \(\mathrm{Cat}_{(\infty, {k})}\).