Local classes

Let \(\Sigma\) be a class of morphisms in a topos \(T\). We say that \(\Sigma\) is a local class if the following conditions are satisfied:

1. It is closed under base change;

2. It is closed under small coproducts;

3. Given a pullback square in \(T\) of the form

   

   in which \(f' \in \Sigma\) and \(g\) is an effective epimorphism, we also have \(f \in \Sigma\).