For any \(n\geq -2, k\geq 0\) and any \((\infty,k)\)-category \(\mathcal C\) and objects \(c,c' \in\mathcal C\), the adjunction ([00AV]) induces an equivalence \[\tau_{n-1} \underline{\mathrm{Hom}}_{\mathcal C}(c,c') \xrightarrow{\simeq} \underline{\mathrm{Hom}}_{\tau_n \mathcal C}(c,c').\]
Proof.
Consider the factorization \(\mathcal C\rightarrow\tau_n \mathcal C\rightarrow{\sf pt}\) into an \(n\)-surjective followed by an \(n\)-faithful functor. Since \(n\)-faithfulness/surjectivity implies homwise \((n-1)\)-faithfulness/surjectivity, it follows that for any \(c,c' \in \mathcal C\), in the induced factorization on hom-\((\infty,k-1)\)-categories \[\underline{\mathrm{Hom}}_{\mathcal C}(c,c') \rightarrow\underline{\mathrm{Hom}}_{\tau_n \mathcal C}(c,c') \rightarrow{\sf pt}\] the first functor is \((n-1)\)-surjective and the second functor is \((n-1)\)-faithful, hence exhibiting \(\underline{\mathrm{Hom}}_{\tau_n \mathcal C}(c,c')\) as the unique factorization \(\tau_{n-1} \underline{\mathrm{Hom}}_{\mathcal C}(c,c')\). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2