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Fibered product


In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) 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 situtations, X ×ZY may intuitively be thought of as consisting of pairs of elements (x,y) with xX and yY and f(x)  =  g(y). 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.

Explicitly, a pullback of the morphisms f and g consists of an object P and two morphisms p1 : P → X and p2 : P → Y for which the diagram

commutes. Moreover, the pullback (P, p1, p2) must be universal with respect to this diagram. That is, for any other such triple (Q, q1, q2) for which the following diagram commutes, there must exist a unique u : Q → P (called a mediating morphism) such that


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