Given a space \(X \in \mathcal S\) and \(k \geq 0\), the map \(\emptyset \rightarrow X\) induces a functor of \((\infty,k)\)-categories \(\partial c_k = \Sigma^k[\emptyset] \rightarrow\Sigma^k [X]\). It then follows from the universal property of \(\Sigma\) that for any \(\mathcal C\in \mathrm{Cat}_{(\infty, {k})}\) and any pair of parallel \((k-1)\)-morphisms \(\alpha\colon \partial c_k \rightarrow\mathcal C\), we obtain an equivalence of spaces \[\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\Sigma^k[X], \mathcal C) \times_{\mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\partial c_k, \mathcal C)} \{\alpha\} \simeq \mathrm{Hom}_{\mathcal S}(X,\mathrm{kHom}_{\mathcal C}(\alpha)).\]
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2