nLab stability under pullback

Idea

There are several contexts in which it is of relevance that a certain property of a morphism $f : A \to B$ is preserved (or stable) under pullback, i.e. also shared by the the morphism $\tilde{f}: X \times_B A \to X$ for any pullback diagram

$$\array{ X \times_B A &\longrightarrow& A \\ {}^{\mathllap{\tilde{f}}} \big\downarrow && \big\downarrow {}^{\mathrlap{f}} \\ X &\longrightarrow& B } \,.$$

Geometers prefer to say “stable under base change”.

Examples

• Monomorphisms are always stable under pullback; that is, if $f$ is a monomorphism, then so is $\tilde{f}$.

• Epimorphisms are not necessarily stable under pullback. In a regular category, the regular epimorphisms are stable under pullback.

• 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.

Dual concept

By duality, a class of morphisms is called stable under pushout (or cobase change) if for every morphism $f : A \to B$ in the class and every morphism $A \to X$ the induced morphism $X \to X \sqcup_A B$ also belongs to the class.

For example, epimorphisms are always stable under pushout, but monomorphisms are not necessarily stable under pushout. A counterexample for the category of groups can be found here; this shows that the category of groups is not coregular?.

Related concepts

• pullback-stable colimit

• stable factorization system

• extensive category