Remark 3.3.
The truncation functor \(\tau_n\colon C \to C_{\leq n}\) always exists when \(C\) is presentable, hence in particular when \(T\) is a topos.
To see this, recall that \(C\) admits a tensoring and a cotensoring by the category \(\An\). There are unique functors
satisfying \(* \otimes X \cong X \cong X^*\) for all \(X \in C\) and preserving colimits (resp. limits) in the first variable. Here and below we use the convention \(S^{-1} := \emptyset\). Thus, for \(n=-2\), the map below is the terminal map \(X \to X^{\emptyset} \cong *\). By an easy induction argument, an object \(X \in C\) is \(n\)-truncated if and only if the map \(X \to X^{S^{n+1}}\) induced by \(S^{n+1} \to *\) is an isomorphism.
Since \(C\) is presentable, there is a small collection of objects \(\{Y\}\) that detect equivalences. Since \(\Hom_C(Y, X^{S^{n+1}}) \simeq \Hom_C(Y \otimes S^{n+1}, X)\), the \(n\)-truncated objects are precisely the local objects with respect to the small collection of morphisms \(\{S^{n+1} \otimes Y \to * \otimes Y = Y\}\). In particular, the \(n\)-truncated objects form a reflective subcategory of \(C\), admitting a reflector \(\tau_n\colon C \to C_{\leq n}\).