Unpacked, the functor \(\tau_n\) can be described as follows, depending on \(n\geq -2\) and \(k \geq 0\).
For any \((\infty,k)\)-category \(\mathcal C\), we have \(\tau_{-2} \mathcal C= {\sf pt}\).
For any \((\infty,k)\)-category \(\mathcal C\), we have \(\tau_{-1} \mathcal C= \emptyset\) if \(\mathcal C\) is empty and \(\tau_{-1} \mathcal C= {\sf pt}\) otherwise.
For \(n \geq k \geq 0\) and any \((\infty,k)\)-category \(\mathcal C\), \(\tau_n \mathcal C\in \mathrm{Cat}_{({n}, {k})}\) is obtained by \((n-k)\)-truncating its \(k\)-morphism spaces (notation 5.1.8), and univalently completing the result. In particular, for an \((\infty,1)\)-category \(\mathcal C\), \(\tau_1\mathcal C\) is its ordinary homotopy category.
For \(k \geq 1\) and any \((\infty,k)\)-category \(\mathcal C\), \(\tau_0 \mathcal C\in \mathrm{Cat}_{({0}, {k})} \simeq \mathrm{Cat}_{({0}, {1})}\) is the posetification of \(\mathcal C\), obtained by applying \(\tau_{-1}\) to its hom-\((\infty,k-1)\)-categories.
For \(k>n \geq 0\) and any \((\infty,k)\)-category \(\mathcal C\), \(\tau_n \mathcal C\in \mathrm{Cat}_{({n}, {k})} \simeq \mathrm{Cat}_{({n}, {n+1})}\) is obtained by \(k\)-homwise applying \(\tau_{-1}\) (and univalently completing the result).