# Contents

## Definition

A section of a morphism $f:A\to B$ in some category is a right-inverse: a morphism $\sigma :B\to A$ such that

$f\circ \sigma :B\stackrel{\sigma }{\to }A\stackrel{f}{\to }B$f \circ \sigma : B \stackrel{\sigma}{\to} A \stackrel{f}{\to} B

equals the identity morphism on $B$.

## Split idempotents

In the case that $f$ has a section $\sigma$, $f$ may also be called a retraction or cosection of $\sigma$, $B$ may be called a retract of $A$, and the entire situation is said to split the idempotent

$A\stackrel{f}{\to }B\stackrel{\sigma }{\to }A\phantom{\rule{thinmathspace}{0ex}}.$A \stackrel{f}{\to} B \stackrel{\sigma}{\to} A \,.

A split epimorphism is a morphism that has a section; a split monomorphism is a morphism that is a section. A split coequalizer is a particular kind of split epimorphism.

## Sections of bundles and sheaves

If one thinks of $f:A\to B$ as a bundle then its sections are sometimes called global sections. This leads to a notion of global sections of sheaves and further of objects in a general topos. See

for more on this.

Revised on January 21, 2013 18:54:45 by Urs Schreiber (89.204.139.229)