Lemma 3.46.
Let \(n \geq 2\), and let \(X\) be an \(n\)-gerbe. Then there exists a unique \(A \in \Ab(T_{\leq 0})\) such that \(\pi_n X \cong X \times A\).
Proof
Since \(X\) is \((n-1)\)-connected, the functor \(X \times -\colon T \to T_{/X}\) induces an equivalence \(T_{\leq n-2} \iso (T_{/X})_{\leq n-2}\) on \((n-2)\)-truncated objects, by Corollary 3.35. In particular, the abelian group object \(\pi_n(X) \in \Ab((T_{/X})_{\leq 0})\) corresponds to an object \(A \in \Ab(T_{\leq 0})\) under this equivalence.