Lemma 2.38.
Any morphism which has a section is an effective epimorphism.
Proof
Assume \(f\colon X \to Y\) admits a section \(s\colon Y \to X\). Then the augmented Čech nerve \(\check{C}^+_{\bullet}(f)\colon \simp\catop_+ \to C\) admits an extra degeneracy, hence is a colimit diagram.