Typically $A \stackrel{f}{\to} B$ is thought of as a bundle and then one speaks of sections of bundles. For topological bundles one considers continuous sections, for smooth bundles smooth sections, etc.

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

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

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