Difference between revisions of "Fiber product"
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== Definition == | == Definition == | ||
| − | Let <math>X</math>, <math>Y</math>, and <math>Z</math> be | + | 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 
, 
, and 
be objects of the same category; let 
and 
be homomorphisms of this category. Then the fiber product of and with respect to , denoted 
(when the specific functions 
and 
are clear) is the set of elements 
in the product 
in which 
.
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