Lemma 3.19.
Consider a pullback square in a topos
in which \(g\) is an effective epimorphism. Then, for every \(n \geq -2\), the morphism \(f\) is \(n\)-truncated (resp. \(n\)-connected) if and only if \(f'\) is \(n\)-truncated (resp. \(n\)-connected).
Proof
Both properties are preserved by arbitrary base change, by Lemma 3.5 and Proposition 3.18. We prove the converses.For truncatedness, argue by induction on \(n\). If \(n=-2\), the claim says that an isomorphism can be detected after pullback along an effective epimorphism, which follows from Lemma 2.29. For \(n\geq -1\), the diagonal of \(f'\) is obtained from the diagonal of \(f\) by base change along the effective epimorphism
\[X'\times_{Y'}X' \longrightarrow X\times_YX.\]
The induction hypothesis therefore shows that \(\Delta_f\) is \((n-1)\)-truncated whenever \(\Delta_{f'}\) is.For connectedness, Corollary 3.10 gives an isomorphism \[\tau_n(X'/Y') \cong g^*\tau_n(X/Y).\]
If \(f'\) is \(n\)-connected, the pullback of \(\tau_n(X/Y)\to Y\) along \(g\) is therefore an isomorphism. Pullback along \(g\) is conservative by Lemma 2.29, so \(\tau_n(X/Y)\to Y\) is itself an isomorphism. Thus \(f\) is \(n\)-connected.