Lemma 5.4.
Every acyclic class in \(T\) is local, in the sense of Definition 2.46. In a modality \((L,R)\) on \(T\), the right class \(R\) is local as well.
Proof
Let \(L\) be an acyclic class, let \(f\colon X \to Z\) be a morphism, and let \(p\colon Z' \to Z\) be an effective epimorphism such that the base change \(f'\colon X \times_Z Z' \to Z'\) lies in \(L\). Form the Čech nerve \(Z'_{\bullet}\to Z\) of \(p\). Every level of the induced morphism
\[X\times_Z Z'_{\bullet}\longrightarrow Z'_{\bullet}\]
is a base change of \(f'\), hence lies in \(L\). Since \(p\) is an effective epimorphism, the colimit of this morphism in \(\Ar(T)\) is \(f\). The class \(L\) is saturated and therefore closed under colimits, so \(f\in L\).Now let \((L,R)\) be a modality and assume instead that \(f'\in R\). Factor \(f\) as \(X\xrightarrow{l}Y\xrightarrow{r}Z\) with \(l\in L\) and \(r\in R\). After base change along \(p\), this is an \((L,R)\)-factorization of \(f'\). Hence the base change of \(l\) is an isomorphism. Since pullback along an effective epimorphism is conservative, \(l\) is an isomorphism and \(f\in R\).