Section of a mapping

$p:X \rightarrow Y$

A mapping $s : Y \rightarrow X$ for which $p \circ s = \mathrm{id}_Y$. In a wider sense, a section of any morphism in an arbitrary category is a right-inverse morphism.

Comments

If $U \subset Y$ is a subset of $Y$, a section over $U$ of $p$ is a mapping $s : U \rightarrow X$ such that $p(s(u)) = u$ for all $u \in U$.

For a vector bundle $E \stackrel{p}{\rightarrow} Y$, where the mapping $p$ is part of the structure defined, one speaks of a section of the vector bundle $E$ rather than of a section of $p$. This applies, e.g., also to sheaves and fibrations. A standard notation for the set of sections in such a case is $\Gamma(E)$, or $\Gamma(U,E)$ for the set of sections of $E$ over $U$.