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Fiber product

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The fiber product, also called the pullback, is an idea in category theory which occurs in many areas of mathematics.

Definition

Let X, Y, and Z be objects of the same category; let \phi : X \to Z and \psi : Y \to Z be homomorphisms of this category. Then the fiber product of X and Y with respect to Z, denoted X \times_Z Y (when the specific functions \phi and \psi are clear) is the set of elements (x,y) in the product X \times Y in which \phi(x) = \psi(y).

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