Let \(n,k \geq 0\), \(\mathcal O\) an \(\infty\)-operad, and \(\mathcal D\) an \(\mathcal O\)-monoidal \((\infty, k)\)-category. Then the \(n\)-homotopy category functor \(h_n\) induces an equivalence of \(\infty\)-categories: \[{(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{(\infty, {k})}))}_{\small{/^{(n-1)}}{\mathcal D}} \xrightarrow{h_n} {(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{({n}, {n})}))}_{\small{/^{(n-1)}}{h_n\mathcal D}}.\]
Proof.
The full subcategory \(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})}) \subseteq \mathrm{Ar}(\mathrm{Cat}_{(\infty, {k})})\) is closed under products and hence defines a Cartesian symmetric monoidal subcategory. Since all functors in ([00BC]) preserve products, the pullback square is a pullback square of Cartesian symmetric monoidal \(\infty\)-categories and hence induces a pullback square of \(\infty\)-categories: 
Under the equivalence of \(\infty\)-categories \(\mathrm{Ar}(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{(\infty, {k})})) \simeq \mathrm{Alg}_{\mathcal O}(\mathrm{Ar}(\mathrm{Cat}_{(\infty, {k})}))\), the full subcategory \(\mathrm{Ar}^{(n-1)}(\mathrm{Alg}_{\mathcal O}(\mathrm{Cat}_{(\infty, {k})}))\) on the \(\mathcal O\)-monoidal functors \(F\) whose underlying functors \(F_X\) are \((n-1)\)-faithful becomes identified with \(\mathrm{Alg}_{\mathcal O}\left(\mathrm{Ar}^{(n-1)}(\mathrm{Cat}_{(\infty, {k})})\right)\). Hence, the pullback square ([00BH]) is equivalent to the square 
and taking fibers at the \(\mathcal O\)-algebra \(\mathcal D\in\mathrm{Alg}_{\mathcal O} \left(\mathrm{Cat}_{(\infty, {k})}\right)\) induces the desired equivalence. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2