Hypercomplete if and only if hypercovers effective

\((1) \Rightarrow (2)\): Let \(U_\bullet\) be a hypercover in \(T_{/X}\), or equivalently a hypercover of \(X\). By Lemma 6.50, the canonical map \(\abs{U_\bullet} \to X\) is \(\infty\)-connected. Since \(T\) is hypercomplete, so is \(T_{/X}\), hence this map is an isomorphism. Thus \(U_\bullet\) is effective.\((2) \Rightarrow (1)\): Let \(f\colon U \to X\) be an \(\infty\)-connected morphism in \(T\). Consider \(f\) as an object of \(T_{/X}\) and let \(f_\bullet\) be the constant simplicial object with value \(f\). By Lemma 6.47, \(f_\bullet\) is a hypercover in \(T_{/X}\). By assumption, \(f_\bullet\) is effective, so its realization is the terminal object \(\id_X\) of \(T_{/X}\). On the other hand, the realization of a constant simplicial object is its constant value, so \(\abs{f_\bullet}\iso f\). Hence \(f\) is an isomorphism, showing that \(T\) is hypercomplete.