Corollary 3.33.
Let \(f\colon X \to Y\) and \(g\colon Y \to Z\) be maps. If \(gf\) is \(n\)-connected and \(g\) is \((n+1)\)-connected, then \(f\) is \(n\)-connected.
Proof
Dual to the proof of Lemma 3.7(2), we may factor \(f\) as \(X \xrightarrow{(\id_X,f)} X \times_Z Y \xrightarrow{\pr_Y} Y\), where the first map is a base change of \(\Delta_g\), hence \(n\)-connected by the theorem, and the second map is a base change of \(gf\).