Lemma 5.61.
A congruence \(K\) is epigenic if and only if it is contained in the class of \(\infty\)-connected maps.
Proof
If \(K\) is contained in \(\Conn_{\infty}\), then it is in particular contained in \(\EffEpi\), so that \(K\) is epigenic. Conversely, assume that \(K \subseteq \EffEpi\). For any morphism \(f\) in \(K\), the iterated diagonals \(\Delta^n_f\) of \(f\) also lie in \(K\), hence are effective epimorphisms by assumption. By Theorem 3.22 it follows inductively that \(f\) is \(n\)-connected for all \(n\geq 0\), hence \(\infty\)-connected.