Pullback

In mathematics, the term pullback is used in two ways that are only loosely related.

• To express the transformation property of a contravariant quantity such as a differential form; for example, given a smooth mapping

f:M → N

of smooth manifolds and a differential form α on N, the resulting differential form

f^*(α)

defined on M is called the pullback of α.

• As a name for a particular kind of inverse limit in category theory, corresponding to a diagram of two morphisms

g:Y → X and h:Z → X

with common target. The resulting limit is called the pullback, or also fiber product. It is often written

Y×_XZ

and in the category of sets is defined as the subset of Y×Z of pairs (y,z) such that

g(y) = h(z).

See also equalizer. The categorical dual notion of colimit is a pushout.