Lemma 6.50.
Let \(T\) be a topos and let \(U_\bullet\) be a hypercover in \(T\). Then the canonical map \(\abs{U_\bullet} \to *\) is \(\infty\)-connected.
Proof
We show that \(\abs{U_\bullet}\) is \(n\)-connected for every \(n \geq 0\). Let \(V_\bullet = \cosk_{n+1} U_\bullet\) and let \(u_\bullet\colon U_\bullet \to V_\bullet\) be the unit map. The matching maps of \(V_\bullet\) agree with those of \(U_\bullet\) through degree \(n+1\) and are isomorphisms thereafter, so \(V_\bullet\) is again a hypercover. Moreover, \(u_m\) is an isomorphism for \(m \leq n+1\). By Lemma 6.49, the induced map \(\abs{U_\bullet} \to \abs{V_\bullet}\) is \(n\)-connected. By Lemma 6.48, \(\abs{V_\bullet} \simeq *\). It follows that \(\abs{U_\bullet}\) is \(n\)-connected.