Corollary 6.46.
Let \(T\) be a topos with hypercompletion \(L\colon T \to T^{\hyp}\). A simplicial object \(U_\bullet\) is a hypercover in \(T\) if and only if \(L(U_\bullet)\) is a hypercover in \(T^{\hyp}\).
Proof
The “only if” direction is Lemma 6.45. Conversely, suppose that \(L(U_\bullet)\) is a hypercover. Since \(L\) is left exact, it preserves the matching objects, so it sends every matching map
\[f_n\colon U_n\longrightarrow(\cosk_{n-1}U_\bullet)_n\]
to an effective epimorphism. Factor \(f_n\) as an effective epimorphism followed by its image monomorphism \(m_n\). The functor \(L\) preserves this factorization. Since \(L(f_n)\) is an effective epimorphism, right cancellation shows that \(L(m_n)\) is an effective epimorphism; being monic, it is an isomorphism. Hence \(m_n\) belongs to the kernel of hypercompletion and is therefore \(\infty\)-connected. Since it is also a monomorphism, it is an isomorphism. Thus \(f_n\) is an effective epimorphism for every \(n\), proving that \(U_\bullet\) is a hypercover.