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.

  • 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, 2015 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, 2015 at 7:03

1 Answer 1


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.

  • $\begingroup$ 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, 2015 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, 2018 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, 2018 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, 2018 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, 2018 at 18:13

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