For \(j > k \geq 0\) and \(\mathcal C\in \mathrm{Cat}_{(\infty, {j})}\), the inclusion of the maximal sub-\((\infty,k)\)-category \(\iota_k \mathcal C\rightarrow\mathcal C\) is \((k-1)\)-surjective.
Proof.
Factoring \(\iota_k \mathcal C\rightarrow\iota_{k+1} \mathcal C\rightarrow\ldots \rightarrow\iota_{j-1} \mathcal C\rightarrow\mathcal C\), it suffices to prove the case \(j=k+1\). For \(k=0\) and \(\mathcal C\in \mathrm{Cat}_{(\infty, {1})}\), the functor \(\iota_0 \mathcal C\rightarrow\mathcal C\) is surjective on objects, i.e. \((-1)\)-surjective. For \(k\geq 1\) and \(\mathcal C\in \mathrm{Cat}_{(\infty, {k+1})}\), the functor \(\iota_k \mathcal C\rightarrow\mathcal C\) is surjective on objects and by induction homwise \((k-2)\)-surjective, hence \(\iota_k \mathcal C\rightarrow\mathcal C\) is \((k-1)\)-surjective. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2