For \(n, k \geq 0\), define the homotopy \(n\)-category functor \[h_n\colon \mathrm{Cat}_{(\infty, {k})} \rightarrow\mathrm{Cat}_{({n}, {n})}\] to be \(\mathrm{Cat}_{(\infty, {k})} \xrightarrow{\tau_n} \mathrm{Cat}_{({n}, {k})} \hookrightarrow \mathrm{Cat}_{({n}, {n})}\) when \(n \geq k\) and to be \(\mathrm{Cat}_{(\infty, {k})} \xrightarrow{\iota_n} \mathrm{Cat}_{(\infty, {n})} \xrightarrow{\tau_n} \mathrm{Cat}_{({n}, {n})}\) when \(n < k\). Note that \(h_n\) is symmetric monoidal as \(\iota_n\) and \(\tau_n\) are symmetric monoidal. For \(\mathcal C\in \mathrm{Cat}_{(\infty, {k})}\), we call \(h_n \mathcal C\) the homotopy \(n\)-category of \(\mathcal C\).
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2