 Section of a morphism

A section of a morphism $p:E\to B$ in a category is a morphism $s:B\to E$ such that $p\circ s=1_B$, where $1_B$ is the identity morphism. Thus a section is a right inverse of $p$; the equation $s\circ p=1_E$ is not required.

For the projection $p:B\times F\to B$, choosing $f_0\in F$ gives $s(b)=(b,f_0)$. Then $p(s(b))=b$, while $s(p(b,f))=(b,f_0)$ may differ from $(b,f)$. This example also explains sections of a fiber bundle: they choose one point in every fibre compatibly with the structure. A section of a morphism is a monomorphism, since $s\circ u=s\circ v$ implies $u=p\circ s\circ u=p\circ s\circ v=v$.