Difference between revisions of "Fiber product"

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== Definition ==
 
== Definition ==
  
Let <math>X</math>, <math>Y</math>, and <math>Z</math> be structures of the same species; let <math>\phi : X \to Z</math> and <math>\psi : Y \to Z</math> be [[homomorphism]]s of this species of structure.  Then the fiber product of <math>X</math> and <math>Y</math> with respect to <math>Z</math>, denoted <math>X \times_Z Y</math> (when the specific functions <math>\phi</math> and <math>\psi</math> are clear) is the set of elements <math>(x,y)</math> in the product <math>X \times Y</math> in which <math>\phi(x) = \psi(y)</math>.
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Let <math>X</math>, <math>Y</math>, and <math>Z</math> be objects of the same category; let <math>\phi : X \to Z</math> and <math>\psi : Y \to Z</math> be [[homomorphism]]s of this category.  Then the fiber product of <math>X</math> and <math>Y</math> with respect to <math>Z</math>, denoted <math>X \times_Z Y</math> (when the specific functions <math>\phi</math> and <math>\psi</math> are clear) is the set of elements <math>(x,y)</math> in the [[Limit (category theory)|product]] <math>X \times Y</math> in which <math>\phi(x) = \psi(y)</math>.
  
 
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Latest revision as of 20:55, 7 September 2008

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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