Proposition 6.108. (Coherence and connectivity)
Consider a morphism \(f\colon X \to Y\).
If \(X\) is \(n\)-coherent and \(f\) is \((n-1)\)-connected, then \(Y\) is \(n\)-coherent.
If \(Y\) is \(n\)-coherent and \(f\) is \(n\)-connected, then \(X\) is \(n\)-coherent.
If \(Y\) is \(n\)-coherent, \(f\) is \((n-2)\)-truncated, and \(X\) is \((n-1)\)-coherent, then \(X\) is \(n\)-coherent.
Proof
We argue simultaneously for the three assertions by induction on \(n\). For \(n=0\), the first assertion says that effective-epimorphic images of quasi-compact objects are quasi-compact, while the second and third reduce to the stability of quasi-compactness under pullback along monomorphisms. Suppose now that the assertions are known in degree \(n-1\). The characterization in Proposition 6.105 reduces \(n\)-coherence to quasi-compactness together with the \((n-1)\)-coherence of pullbacks against a covering family of \((n-1)\)-coherent objects. After pulling back \(f\) along such a family, its connectivity or truncation bound is unchanged. The three required statements about these pullbacks are then exactly the induction hypotheses in the relevant slice topoi. This proves all three assertions.