Definition 3.1.
Let \(C\) be a category with pullbacks. A morphism \(f\colon X \to Y\) in \(C\) is called \((-2)\)-truncated if it is an isomorphism. For \(n \geq -1\), we say \(f\) is \(n\)-truncated if \(\Delta_f\colon X \to X \times_Y X\) is \((n-1)\)-truncated. We denote by \(C_{\leq n} \subseteq C\) the full subcategory of \(n\)-truncated objects.
If the inclusion \(C_{\leq n} \hookrightarrow C\) admits a left adjoint, we denote this adjoint by
\[\tau_n\colon C \to C_{\leq n},\]
and call it the \(n\)-truncation functor.