Geometers prefer to say “stable under base change”.
Monomorphisms are always stable under pullback; that is, if is a monomorphism, then so is .
The right lifting property: Generally, the property of a morphism of having a right lifting property is stable under pullback. Therefore for instance fibrations and acyclic fibrations in a model category are stable under pullback. If also weak equivalences are stable under pullback along fibrations, then one speaks of a right proper model category.
Similarly, the property of being right orthogonal to a class of morphisms is stable under pullback. Thus, the right class in any orthogonal factorization system is stable under pullback. If the left class is also pullback-stable, the OFS is called stable.