In Conceptual Mathematics, Lawvere and Schanuel define a section (given $f: A \rightarrow B$), as a map $s: B \rightarrow A$ such that $f \circ s = 1_B$.
(A retraction in their terminology is a map $r: B \rightarrow A$ such that $r \circ f = 1_B$.)
I'm wondering about the meaning and source of these terms. I haven't seen them much if anywhere else in the literature. (Admittedly I'm a novice.) Are they simply convenient terms for a simple kind of map, or are they meant to resonate with topological or topos-theoretical concepts?
The following is from Sheaves in Geometry and Logic by Mac Lane and Moerdijk:
For each open set $U \subset X$ one can then consider the "sections" s over $U$ of the sheaf $A$: each section is a function which selects—again in a suitably continuous way—for each point $x \in U$ an element $s(x)$ in the corresponding abelian group $A_x$. (page 2)
Do these "sections" have features in common, or do they just happen to share the term?
As far as I know, both words, "retraction" and "section", first acquired these meanings in topology.
The fundamental example of a retraction arises in the context of a subspace $X$ of a space $Y$, when $r:Y\to X$ is a continuous map that fixes all points of $X$. Consider the case where $X$ is the unit disk and $Y$ is the plane containing it. Let $r$ map each point in $X$ to itself and map each point of $Y-X$ to the point on the unit circle with the same angular coordinate. So all points in $Y$ outside of $X$ are being pulled back, radially toward the origin, until they arrive at $X$. (Note the etymology of "retract" form "back" and "pull".)
The general case of a retraction $r:Y\to X$ is similar except for two points. First, in general, $X$ need not be literally a subspace of $Y$; it's enough that it be homeomorphic to a subspace $X'$ of $Y$ (by a homeomorphism under which $r$ corresponds to a retraction in the sense described above). Second, my example of the circle and the plane is special in that one can continuously deform the points in $Y$ toward their images in $X$, just by moving them radially. The general notion of retraction doesn't require such a deformation (homotopy). If one exists, one speaks of a deformation retract; if all points in $X$ remain fixed during the whole deformation, then one has a strong deformation retract.
The mental picture for "section" is to consider the projection of the plane $\mathbb R^2$ to the $x$-axis. A section $s:\mathbb R\to\mathbb R^2$ of this $f$ is a function assigning to each point on the $x$-axis a point in the fiber (vertical line) at $x$. Visually, it cuts through all those fibers. (Again, note the etymology of "section" from a Latin word meaning cutting.)