Corollary 3.34.
Let \(n\geq0\), and consider a \(0\)-connected pointed object \(X \in T_*^{\geq 1}\). Then \(X\) is \(n\)-connected if and only if \(\Omega X\) is \((n-1)\)-connected. Similarly, \(X\) is \(n\)-truncated if and only if \(\Omega X\) is \((n-1)\)-truncated.
Proof
By the theorem, \(X\) is \(n\)-connected if and only if \(\Delta\colon X \to X \times X\) is \((n-1)\)-connected. Given a base point \(x\colon * \to X\), it follows from \(0\)-connectedness and Corollary 3.33 that \(x\) is \((-1)\)-connected. Hence \((x,x)\colon * \to X \times X\) is an effective epimorphism. By Lemma 3.19, the diagonal \(\Delta\) is \((n-1)\)-connected if and only if its base change \(\Omega X \to *\) is \((n-1)\)-connected.For truncatedness, \(X\) is \(n\)-truncated if and only if \(\Delta\) is \((n-1)\)-truncated. The same locality lemma identifies this with \((n-1)\)-truncatedness of \(\Omega X\to *\).