nLab
section

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

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

equals the identity morphism on $B$.

Split idempotents

In the case 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 \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 fork diagram

$$A\underset{{\partial }_0}{\overset{{\partial }_1}{\rightrightarrows }}B\stackrel{p}{\to }C$$A \underoverset{\partial_0}{\partial_1}{\rightrightarrows} B \overset{p}{\rightarrow} C

is a split fork if the morphism $p$ is a retract, in the category of arrows, of the morphism ${\partial }_1$, via a section of the pair $({\partial }_0,p):{\partial }_1\to p$. In other words, there exist a pair (section) $(t,s):p\to {\partial }_1$ of $({\partial }_0,p)$ such that the diagram

$$\begin{array}{ccccc}B& \stackrel{t}{\to }& A& \stackrel{{\partial }_0}{\to }& B\\ \downarrow p& & \downarrow {\partial }_1& & \downarrow p\\ C& \stackrel{s}{\to }& B& \stackrel{p}{\to }& C\end{array}$$\array{B &\stackrel{t}\rightarrow & A&\stackrel{\partial_0}\rightarrow &B\\ \downarrow p &&\downarrow {\partial_1} &&\downarrow p\\ C&\stackrel{s}\rightarrow & B&\stackrel{p}\rightarrow& C }

commutes and its both horizontal composites are identities. It is easy to check that every split fork is a coequalizer, which is then called a split coequalizer. Clearly any functor sends a split coequalizer to a split coequalizer, hence every split coequalizer is an absolute coequalizer? (coequalizer stable under all functors).

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

• global section

for more on this.