Let \(k, n \geq 0\) and let \(F \colon \mathcal C\rightarrow\mathcal D\) be an \((n-1)\)-faithful functor between \((\infty,k)\)-categories. Then, for every \(\mathcal X\in \mathrm{Cat}_{(\infty, {k})}\), the square 
is a pullback square of spaces.
Equivalently, \(F\) is a Cartesian morphism for the functor \(h_n \colon \mathrm{Cat}_{(\infty, {k})} \rightarrow\mathrm{Cat}_{({n}, {n})}\).
Proof.
For any pair of objects in \(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\), the pullback square ([00BC]) of \(\infty\)-categories induces a pullback square between the respective hom-spaces. In particular, for the pair \((\mathrm{id}_{\mathcal X} \colon \mathcal X\rightarrow\mathcal X)\) and (\(F \colon \mathcal C\rightarrow\mathcal D\)) of objects in \(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\), we note that \[\mathrm{Hom}_{\mathrm{Ar}(\mathrm{Cat}_{(\infty, {k})})}(\mathrm{id}_{\mathcal X}, F) \simeq \mathrm{Hom}_{\mathrm{Cat}_{(\infty, {k})}}(\mathcal X, \mathcal C) \quad \mathrm{Hom}_{ \mathrm{Ar}(\mathrm{Cat}_{({n}, {n})})}(\mathrm{id}_{h_k \mathcal X}, h_k F) \simeq \mathrm{Hom}_{\mathrm{Cat}_{({n}, {n})}}(h_k\mathcal X, h_k\mathcal C),\] and hence that the resulting pullback square of hom-spaces precisely results in the square ([00BJ]). ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2