7
$\begingroup$

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.

$\endgroup$
2
  • 4
    $\begingroup$ I think the terminology is inspired from topology. A retraction in topology is special case of the category retraction and does retract a topological space to a subspace. Section terminology on the other hand is inspired from pretty much any sheaf like structure (covering space, fiber bundle, sheaf, etc.) which is again a special case of the category section. $\endgroup$
    – Hamed
    Nov 2 '15 at 5:40
  • $\begingroup$ Notice that $e$ follows $s$ so it is just like a retracted story. The morphism $s$ takes $A$ to $X$ and the retraction $e$ brings it back to extactly as it was. $\endgroup$
    – John Douma
    Nov 2 '15 at 7:03
11
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ Typo: in the definition of deformation retract, it should be $r i \simeq \mathrm{id}_A$. (Also, if you want the 'shrink X onto A' feeling, maybe strong deformation retracts are best candidates.) $\endgroup$
    – Pece
    Nov 2 '15 at 6:48
  • $\begingroup$ @Hamed, when you say This is called a section of the vector bundle for obvious reasons , are you saying obvious in the sense that it is similar with the use of the word section that we use in everyday English? If so, how? $\endgroup$
    – roi_saumon
    Oct 10 '18 at 14:04
  • 1
    $\begingroup$ @roi_saumon Let's go with an easier to understand example than a vector bundle. Consider the trivial fiber bundle $X\times \{0,1\}\to X$, i.e. $X$ is some base space and fiber has only two elements (the map is the usual projection). The space $E=X\times \{0,1\}$ is known as the total space. A section of $E$, in the English sense of the word, would be a group of points of $E$ that have some property. Like section in the example "the children's section of the library".... $\endgroup$
    – Hamed
    Oct 10 '18 at 18:06
  • 1
    $\begingroup$ ... Now let's understand a point of $E$ as a tuple $(x, b)$ where $x\in X$ and $b\in \{0,1\}$. The only sensible way, for mathematics, to group certain points of $E$ and call it a section is if for every $x\in X$ we choose either $(x,0)$ or $(x,1)$ to be part of the group. In other words, each point $x\in X$ needs to have one and only one representative in the section... $\endgroup$
    – Hamed
    Oct 10 '18 at 18:10
  • 1
    $\begingroup$ To generalize to a vector bundle over base $X$ now, you can think of a section as a vector field over $X$. But this is one of many possibilities of vector fields over $X$, so it's only an English-section of the whole story (i.e. the total space $E$). $\endgroup$
    – Hamed
    Oct 10 '18 at 18:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.