Definition 3.41.
Given \(n \geq 0\), we say an object \(X \in T\) is an \(n\)-gerbe if it is \(n\)-truncated and \((n-1)\)-connected. We define an Eilenberg–MacLane object of degree \(n\) to be a pointed \(n\)-gerbe. We denote by
\[\mathrm{Gerb}_n(T) \subseteq T \qquadtext{ and } \mathrm{EM}_n(T) \subseteq T_*\]
the resulting full subcategories. More generally, we refer to a morphism \(X \to Y\) as an \(n\)-gerbe if it is an \(n\)-gerbe in the slice topos \(T_{/Y}\).