Difference between revisions of "Fiber product"
(New page: The '''fiber product''', also called the '''pullback''' is an idea in category theory which occurs in many areas of mathematics. == Definition == Let <math>X</math>, <math>Y</math>, ...) |
m (Better to use category theory terminology) |
||
| (One intermediate revision by one other user not shown) | |||
| Line 1: | Line 1: | ||
| − | The '''fiber product''', also called the '''pullback''' is an idea in [[category theory]] which occurs in many areas of mathematics. | + | The '''fiber product''', also called the '''pullback''', is an idea in [[category theory]] which occurs in many areas of mathematics. |
== 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>. |
{{stub}} | {{stub}} | ||
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 
.
This article is a stub. Help us out by expanding it.
