hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
A section of a morphism $f : A \to B$ in some category is a right-inverse: a morphism $\sigma : B \to A$ such that
equals the identity morphism on $B$.
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
This is the space of sections of $f$. A single section $\sigma$ is a global element in here
See at dependent product – In terms of spaces of sections for more on this.
In the 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 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.
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
for more on this.