nLab section

Definition

Explicitly

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 $B$.

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 terms of dependent product

In a locally cartesian closed category $\mathcal{C}$, regard the morphism $f\colon A \to B$ as an object $[f] \in \mathcal{C}_{/B}$ in the slice category over $B$. Then there is the dependent product

$$\underset{B}{\prod} [f] \in \mathcal{C} \,.$$

This is the space of sections of $f$. A single section $\sigma$ is a global element in here

$$\sigma \colon \ast \to \underset{B}{\prod} [f] \,.$$

See at dependent product – In terms of spaces of sections for more on this.

Split idempotents

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

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

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

• global section

for more on this.

Related concepts

• local section, global section

• flat section

• space of sections

• ex-space

• field history