Coskeletal hypercover effective

The details of the proof are intricate; we refer to [Lurie 2009, Lemma 6.5.3.9] for a detailed argument. We only sketch the idea of Lurie's proof.We proceed by induction on \(n\). If \(n = 0\), then \(U_\bullet\) can be identified with (the underlying groupoid object of) the Čech nerve of the map \(U_0 \to *\). Since \(U_\bullet\) is a hypercover, this map is an effective epimorphism, so the Čech nerve is a colimit diagram and \(\abs{U_\bullet} \simeq *\).Now assume \(n > 0\) and let \(V_\bullet = \cosk_{n-1} U_\bullet\). This is an \((n-1)\)-coskeletal hypercover, so \(\abs{V_\bullet}\simeq *\) by induction. The unit \(f_\bullet\colon U_\bullet \to V_\bullet\) is an isomorphism below degree \(n\). Moreover, each \(f_m\) is a finite composite of pullbacks of the matching map \(f_n\), and hence is an effective epimorphism.Form the degreewise Čech nerve \(W^+\) of \(f_\bullet\), regarded as an augmented bisimplicial object. Realizing in the Čech direction gives \(V_\bullet\), and then realizing in the other direction gives \(\abs{V_\bullet}\simeq *\). By cofinality of the diagonal, the diagonal simplicial object \(D_\bullet\) of the underlying bisimplicial object therefore has terminal realization. Lurie constructs a retract of the underlying semisimplicial object of \(D_\bullet\) onto that of \(U_\bullet\). Forgetting degeneracies does not change geometric realizations, so \(\abs{U_\bullet}\) is a retract of \(\abs{D_\bullet}\simeq *\) and is therefore terminal.