Hypercompletion preserves the monogenic part

For the first part, consider the sequence of localizations \(T \to T/K \to (T/K)^{\hyp}\). Let \(f\) be a monomorphism in \(T\) contained in \(K^{\hyp}\); we need to show that \(f\) is in fact contained in \(K\). Since \(T \to T/K\) is left exact, the image of \(f\) in \(T/K\) is a monomorphism that becomes an equivalence in the hypercompletion \((T/K)^{\hyp}\). Thus, in \(T/K\), the map \(f\) is \(\infty\)-connected. But a map that is both a monomorphism and \(\infty\)-connected is an equivalence. Therefore, \(f\) is already inverted in \(T/K\), so \(f \in K\).For the second part, we always have \((K^{\mono})^{\hyp} \subseteq K^{\hyp}\). For the converse, we may consider the monogenic-epigenic factorization of the quotient \(T \to T/K\): The second map is an epigenic quotient by Proposition 5.64, hence its kernel is contained in the \(\infty\)-connected maps. An epigenic quotient induces an equivalence on hypercompletions, see [Anel et al. 2025, Lemma 2.1.33]. We therefore obtain It follows that \((K^{\mono})^{\hyp} = K^{\hyp}\), as desired.