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In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse. In other words, if $fcolon\; Xto\; Y$ and $gcolon\; Yto\; X$ are morphisms whose composition $fgcolon\; Yto\; Y$ is the identity morphism on Y, then g is a section of f, and f is a retraction of g.## Examples

Given a quotient space $bar\; X$ with quotient map $picolon\; X\; to\; bar\; X$, a section of $pi$ is called a transversal.
## See also

The categorical concept of a section is important in homological algebra, and is also closely related to the notion of a section of a fiber bundle in topology: in the latter case, a section of a fiber bundle is a section of the bundle projection map of the fiber bundle.

Every section is a monomorphism, and every retraction is an epimorphism; they are called respectively a split monomorphism and a split epimorphism (the inverse is the splitting).

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Last updated on Thursday July 10, 2008 at 15:55:14 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Thursday July 10, 2008 at 15:55:14 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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