Pullback (category theory)

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 written

P = X ×_{f, Z, g} Y.

Usually the morphisms f and g are omitted from the notation, and then the pullback is written

P = X ×_Z Y.

The pullback 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 (x, y) with x in X, y in Y, 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.