nLab
pullback stability

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

$$\begin{array}{ccc}X{\times }_{B}A& \to & A\\ {}_{\tilde{f}}\downarrow & & {\downarrow }_f\\ X& \to & B\end{array}.$$\array{ X \times_B A &\to& A \\ _{\tilde{f}}\downarrow && \downarrow_f \\ X &\to& 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 $\tilde{f}$.

• 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 then one speaks of a left proper model category.