We list a few edge cases of Definitions 5.4.1.
For any \(k \geq 0\), there is only one \((-2,k)\)-category, namely \({\sf pt}\).
For any \(k \geq 0\), there are only two \((-1,k)\)-categories, namely \(\emptyset\) and \({\sf pt}\).
For any \(k > 0\), a \((0,k)\)-category is precisely a partially ordered set (i.e. an \(\infty\)-category enriched in \((-1)\)-truncated spaces). In particular, the inclusions \(\mathrm{Cat}_{(0,1)} \hookrightarrow\mathrm{Cat}_{(0,2)} \hookrightarrow\cdots\) are all equivalences.
More generally, for any \(k > n \geq 0\), an \((n,k)\)-category is precisely an \((n+1,n+1)\)-category whose spaces of \((n+1)\)-morphisms are all either empty or contractible. In particular, the inclusions \(\mathrm{Cat}_{(n,n+1)} \hookrightarrow\mathrm{Cat}_{(n,n+2)} \hookrightarrow\cdots\) are all equivalences.
Taking \(k = 1\), for any \(n \geq 1\), an \((n,1)\)-category is precisely an \((\infty,1)\)-category whose hom-spaces are \((n-1)\)-truncated.