There are several contexts in which it is of relevance that a certain property of a morphism is preserved (or stable) under pullback, i.e. also shared by the the morphism for any pullback diagram
Geometers prefer to say “stable under base change”.
Monomorphisms are always stable under pullback; that is, if is a monomorphism, then so is .
In many important kinds of categories; some or all colimits are stable under pullback; this is discussed at commutativity of limits and colimits.
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.
Last revised on May 8, 2022 at 13:46:23. See the history of this page for a list of all contributions to it.