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In category theory, a branch of mathematics, a pullback is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written
and comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, X ×Z Y may intuitively be thought of as consisting of pairs of elements with x in X, y in Y, and f = g. For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.
The dual concept of the pullback is the pushout.
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