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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