category theory

mapping space

# 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

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

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

$A \stackrel{f}{\to} B \stackrel{\sigma}{\to} A \,.$

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 August 22, 2013 05:12:40 by Urs Schreiber (150.212.99.10)