Definition 6.42. (Hypercover)
Let \(T\) be a topos. A simplicial object \(U_{\bullet} \in \Fun(\simp\catop,T)\) is called a hypercover in \(T\) if, for each \(n \geq 0\), the canonical map
\[U_n \to (\cosk_{n-1} U_{\bullet})_n\]
is an effective epimorphism. We say that \(U_{\bullet}\) is an effective hypercover if the colimit of \(U_{\bullet}\) is a terminal object of \(T\).
For an object \(X \in T\), a hypercover of \(X\) is a hypercover in the slice topos \(T_{/X}\).