Sections, Retractions, and Idempotents

Section-Retraction Pairs

Definition

A section-retraction pair is a pair of functions $s : B \to A$ and $r : A \to B$ such that:

$r \circ s = \text{id}_B$

• $s$ is called the section (or left inverse)
• $r$ is called the retraction (or right inverse)

Intuition

• $s$ embeds $B$ into $A$
• $r$ projects back from $A$ to $B$
• Composing $r$ after $s$ returns to where we started in $B$

Examples

• $B \hookrightarrow A \twoheadrightarrow B$ where $s$ is inclusion and $r$ is identity on $B$
• $s : \mathbb{N} \to \mathbb{Z}$, $s(n) = n$; $r : \mathbb{Z} \to \mathbb{N}$, $r(n) = |n|$ fails since $r(s(-1))$ is undefined
• Correct: $s : \mathbb{N} \to \mathbb{Z}$, $s(n) = n$; $r : \mathbb{Z} \to \mathbb{N}$, $r(n) = \max(n, 0)$

Properties

Section is Injective

If $s$ is a section of $r$, then $s$ is injective.

Proof: If $s(b_1) = s(b_2)$, then $b_1 = r(s(b_1)) = r(s(b_2)) = b_2$.

Retraction is Surjective

If $r$ is a retraction of $s$, then $r$ is surjective.

Proof: For any $b \in B$, $b = r(s(b))$, so $b$ is in the image of $r$.

Converse Requires Choice

If $s : B \to A$ is injective, there exists a retraction $r : A \to B$ (on image of $s$) iff we can “pick” values for elements outside the image.

This requires the Axiom of Choice in general.

Idempotents

Definition

A function $e : A \to A$ is idempotent if $e \circ e = e$.

Intuition

Applying $e$ once or twice gives the same result. $e$ “freezes” elements to a fixed form.

Examples

• Projection $(x, y) \mapsto (x, 0)$ is idempotent
• Identity is idempotent
• Constant function $x \mapsto c$ is idempotent iff $c = c$ (always true)

Examinable Proof: Idempotents and Section-Retraction Pairs

Theorem: There is a bijection between:

• Section-retraction pairs $(s, r)$ with $s : B \to A$, $r : A \to B$, $r \circ s = \text{id}_B$
• Idempotents $e : A \to A$

From Section-Retraction to Idempotent

Given $(s, r)$ with $r \circ s = \text{id}_B$, define $e = s \circ r : A \to A$.

Idempotence:

$e \circ e = s \circ r \circ s \circ r = s \circ \text{id}_B \circ r = s \circ r = e$

From Idempotent to Section-Retraction

Given idempotent $e : A \to A$, define:

• $B = \text{Im}(e) = \{e(a) : a \in A\}$
• $r : A \to B$ by $r(a) = e(a)$
• $s : B \hookrightarrow A$ the inclusion

Verification: $r(s(b)) = r(b) = e(b)$.

If $b \in B$, then $b = e(a)$ for some $a$. So $e(b) = e(e(a)) = e(a) = b$ by idempotence.

Thus $r \circ s = \text{id}_B$.

These are Inverses

Starting with $(s, r)$: the associated idempotent $s \circ r$ has image isomorphic to $B$ (via $s$).

Starting with $e$: the section-retraction pair recovers $e$ as $s \circ r$.

Splitting Idempotents

The process of going from $e$ to $(s, r)$ is called splitting the idempotent.

The set $B = \text{Im}(e)$ is the “fixed points” of $e$, and $e$ is the identity on $B$.

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Summary

• Section-retraction: $r \circ s = \text{id}_B$
• Sections are injective; retractions are surjective
• Idempotent: $e \circ e = e$
• Bijection: $(s, r) \mapsto s \circ r$ and $e \mapsto (\text{inclusion}, e|_{\text{Im}(e)})$