Pullback

Summary

The pullback, or fibre product, of two morphisms $f \colon X \to Z$ and $g \colon Y \to Z$ with a common target is a universal commutative square over them. It is the limit of the diagram $X \to Z \leftarrow Y$. In sets it is the set of pairs $(x, y)$ with $f(x) = g(y)$, and it includes preimages and intersections as special cases. Pullbacks are used to restrict, intersect and change the base of objects throughout geometry and topology.