Best way to conceptualize the section/retraction duality? I'm trying to get a feeling for sections and retractions geometrically, so I have the following picture in mind:

In the first vertical, we have a bundle $E \xrightarrow{\pi} B$ where $B$ is a base space and $E$ is the bundle. Then, in this case, we cut the bundle along the pink line, to get a section  $s: B \hookrightarrow E$ a section iff $\pi(s(b)) = b$.
Dually, if we start with a section, only this time written as $i: B \hookrightarrow E$, where we think of the section as giving us an embedding $i$ of $B$ in $E$, then the retract $r: E \rightarrow B$ is obtained by "pushing down" / "retracting" each of the fibers into the section.
However, algebraically speaking, if we have two maps:
$$
\alpha: A \hookrightarrow B \\
\beta: B \twoheadrightarrow A
$$
where $\alpha$ is injective, $\beta$ is surjective, can I say:

*

*$\alpha$ is a section of $\beta$?

*$\beta$ is a retract of $\alpha$?

Is the "two pictures" I have drawn above a faithful representation of how one "ought to think" about sections and retractions?
 A: I doubt there's a "best way" to think of anything in math.
For what it's worth, though, what you've said is almost exactly how I personally think of sections and retracts.
This perspective kind of takes sections as primitive, in that we're thinking of a fibre bundle, and a retract is a retraction of the total space onto the image of the base space under some section. We can instead take retractions as primitive. In that setting, we would have an image like this in mind:

Now the section is simply the inclusion map $\iota : S^1 \to S^1 \times S^1$, and we're retracting onto this subspace. Oftentimes I default to this interpretation when I'm actually dealing with a deformation retract (of course, the above image is not a deformation retract. That's why I chose it).
If you want a purely algebraic condition, whenever you ever have a (possibly one sided) inverse pair $rs = 1$ you can call $r$ a retract of $s$ and $s$ a section of $r$. If you like category theoretic language, a retract is a split epi and a section is a split mono. There's actually a wikipedia article about this, if you haven't seen it yet.

I hope this helps ^_^
