Definition 5.5. (Cotruncation)
We say that a map \(f\colon X \to Y\) is \(n\)-cotruncated if it is \(n\)-truncated in \(T\catop\). Inductively, this equivalently means that the \((-2)\)-cotruncated maps are precisely the isomorphisms, and that a map \(f\) is \((n+1)\)-cotruncated if and only if its codiagonal \(\nabla_f\colon Y \sqcup_X Y \to Y\) is \(n\)-cotruncated.