I wonder what is the intuition behind the definitions of a section and retraction in Category Theory.
From Awodey's book:
Definition 2.7. A split mono (epi) is an arrow with a left (right) inverse. Given arrows $e : X → A$ and $s : A → X$ such that $es = 1_A$, the arrow $s$ is called a section or splitting of $e$, and the arrow $e$ is called a retraction of $s$. The object $A$ is called a retract of $X$.
Of course the names we attribute to the properties are completely arbitrary, but still I wonder why we call $e$ a "retraction" of $s$ here. Does it really "retract" $s$ in any intuitive sense I can't see? Maybe an allusion to a retraction in algebraic topology? What about sections and splits mono/epi then?
I can only see the abstract picture of it, it would be great if someone could shed some light on the intuitions behind those definitions.
Expanding on my comment, let me inspire further:
Retraction Suppose you have a topological space $X$ and $A\subset X$ is a subspace. Hence there is a natural injection map (inclusion) $\imath:A\hookrightarrow X$. We say the continuous map $r:X\to A$ is a retraction map if $r|_{A}=\mathrm{id}$ and $r(X)=A$. In other words although $r$ doesn't touch $A$, it retracts $X$ to $A$. This $r$ has the property that $r\circ\imath=\mathrm{id}_A$. As you see this is pretty much the same properties in your category language (except for it being in the category of topological spaces, $\mathrm{Top}$).
(Remark) There is also a notion of deformation retract: Say you have a continuous map $F:X\times[0,1]\to X$ such that $F(x,0)=x$, $F(x,1)\in A$ and $F(a,1)=a$ for all $t\in [0,1]$ and $a\in A$. Then basically you have a homotopy between the identity map $\mathrm{id}_X$ and a retraction $r$. So not only we have $r\circ\imath=\mathrm{id}_A$ we also have $\imath\circ r\simeq \mathrm{id}_X$. We say the deformation retraction is strong if moreover $F(a,t)=a$ for all $t$. One verbally expects from the words retraction to shrink $X$ to $A$ continuously while not touching $A$ at any given moment; well this is exactly what strong deformation retraction does (Caution: not every retraction is a obtained from a deformation retract! However the intuition helps.)
Section Now let's do it in the other direction. Suppose you have natural surjective map $\pi: E\twoheadrightarrow X$, a projection. This can represent anything from a sheaf, fiber bundle, covering space, etc. For simplicity suppose this represents a vector bundle over a smooth manifold $X$. $E$ is called the total space and for any $x\in X$, $F_x=\pi^{-1}(x)$ is called the fiber over $x$ which is a vector space (hence the name vector bundle). Intuitively on top of every point we put a vector space and glue it all in an appropriate way.
Now suppose you want to define a vector field, i.e. you want to assign to every point $x\in X$ a vector in the fiber $F_x=\pi^{-1}(x)$ in a continuous way. This is essentially a map $\Phi: X\to E$. But since we assign to $x\in X$ a vector in $F_x$, we have the property that $\pi\circ \Phi = \mathrm{id}_X$. This is called a section of the vector bundle for obvious reasons. Again this is exactly the same definition as your category version (except for category being specific).
This is the inspiration behind why $e$ and $s$ (in your definition) are called retractions and sections as general arrows. Note in first example not only $r$ is a retraction, $\imath$ is also a section. And in second example while $\Phi$ is a section, $\pi$ is a retraction. Sections and retractions always come in pairs. Although usually one of them bears more information than the other (if there is a natural injection, retraction is the non-trivial map, and if there is a natural surjection, section is the non-trivial one).
In general the intuition behind a retraction is shrinking a bigger space to a small object. While a section is cutting through the bigger space by means of the smaller space. Hope this helps.