Are these examples of retractions and sections correct?

Question

I asked this on mathoverflow, but I was told to ask this here!

In the book “Conceptual Mathematics: A first introduction to categories” (first edition) on page 52 we can see the following:

Then, on page 53 we have that:

Both ‘mononorphism’ and ‘epimorphism’ are ‘cancellation’ properties.
When we are given both $f$ and $r$, and $r \circ f = 1_A$ then, of course, we can say both that $r$ is a retraction for $f$ and that $f$ is a section for $r$. For which sets $A$ and $B$ can such pairs of maps exist? As we will see more precisely later, it means roughly (for non-empty $A$) that $A$ is smaller (or equal to) in size than $B$.

I find it a bit hard to backtrack everything that is needed to know to get that $|A|≤|B|$, but I'm trying to. My understanding is that if otherwise, $|A|>|B|$, there could be two maps $x_1$ and $x_2$ from $T$ to $A$ that are different ($x_1 \neq x_2$) even though $f(x_1) = f(x_2)$. To properly imagine this, I believe that the following two examples apply:

1. (case $|A|>|B|$) Two male and female (sex, $T$) individuals among all people ($A$) can be different even though they live in the same country ($B$), therefore there’s no retraction for $f$. Because there are even more maps from $A$ to $B$ than the amount of maps from $T$ to $A, each single man belongs in a country at least.

2. (Where $|A| ≤ |B|$) Any selection of religions $x_1$ and $x_2$ (of $T$) from a country ($A$) have only one representative ($B$). Therefore, any two references of a representative of a country $f(x_1)$ and $f(x_2)$ are the same person, which matches the original selection of religions (even if only one).

Are these examples correct? Otherwise, how could they be fixed? (And can you provide better examples?).

Answer 1

In response to @sadasant's answer (Perhaps like a feedback). This answer clarifies that answer, and attempts to provide more clear definitions of concepts and examples. Following are my notes about your answer:

• Your definition of retraction and section appears to be correct. (Reference used: Lawvere and Schanuel’s book Conceptual mathematics 2nd ed.)
• Uniqueness Theorem says that if a morphism $f$ has section $s$ and retraction $r$, then r=s. (Not $r=f$). When you wrote the statement that “If $f$ necessarily has a retraction $r$, the only possible section for $f$ is $r$ itself.” Why did you only say that if $f$ has a retraction r, what about if $f$ has a section? Then also, that section is the retraction. What do you mean by “necessarily”?
• Then you compare $A≥B$, I assume that you mean $cardinality(A)≥cardinality(B)$, where cardinality tells us number of elements in a set. I think, most people may guess what are you trying to say, but it is important to communicate precisely, in case you do not have common base knowledge with the reader. In short, you can say $|A|≥|B|$.
• In the first example, “All people in $A$ live in country $B$”. I thought $B$ is not a country, but set of countries (Happens to contain only 1 country). Therefore, more precise statement will be: All people $p$ in $A$ live in country $b$ in $B$.
• Again, “The country $B$ has a representative in $A$.” It is correct to write that country $b$ in $B$ has only 1 representative in $A$. Representative, like a president of The United States of America.
• “However, there's a retraction between the maps $A$ and $B$”. But $A$ and $B$ are defined to be sets, now you are calling them maps. I am not sure why.
• What do you mean when you say, “necessarily $f$” in 7th point of your answer section “In the first example”.
• Next point that is (i.e.) 8th point under “In the first example” heading, I think you are trying to define monomorphism. Following is one definition of monomorphism: If $f:A→B$. Where $A$ and $B$ are two objects in category $C$. Consider another object $T$ in category $C$, and any pair of maps $x_1,x_2:T→A$. If the statement $S$:If $f∘x_1=f∘x_2$, then $x_1=x_2$ (Note that you wrote $f(x_1)$ to compose maps which is confusing, as $f$ of $x$, where $x$ is an element of set, is also written as $f(x_1)$)is true for any object $T$ and any pair of maps $x_1,x_2$, then $f$ is said to be a monomorphism.
• In last point under heading “On the second example”, I am not sure how to interpret $T$, what do you mean when you say “Criteria defining $A$, called $T$”. I will just give you hint, understand definition of monomorphism and epimorphism.
• Notes regarding your last paragraph: Every object has at least 1 idempotent map namely identity map as $id_A∘id_A=id_A$. Therefore, if there is an object, there is at least 1 idempotent map associated with that object. An example of a map where there is no retraction to a map: suppose $A$ is a set of boys and $B$ is a set of girls, and $|A|>|B|$. A map $Likes:A \rightarrow B$ maps each boy in $A$ to a girl in $B$. Note that, there are at least 2 boys, that are mapped to same girl, let’s say her name is $b$. Now, if any retraction exists, it must map that girl to two boys, which is not possible (By definition of a function), hence retraction cannot exist.

Answer 2

I’m answering myself because I think I learned the correct meaning of sections and retractions. I wasn’t understanding them properly. I will first define section and retraction and later I will challenge the exercises used in the original question.

Definitions:
For a morphism f from two undefined objects A and B: f: A → B

There's a section s: B → A iff f • s = 1B. There's a retraction r: B → A iff r • f = 1A.

The Theorem of Uniqueness says:
If s is a section for f, and r is a retraction for f, them r = f.

In other words:
If f necessarily has a retraction r, the only possible section for f is r itself.

For two sets A and B where A ≥ B:
If A has all its elements mapped to B, so that f:A→B, and all elements of B are mapped to A, so that g:B→A, “g” is a section of “f” if (and only if) B elements remain intact (identity) after all the results of g:B→A are mapped back to B through f:A→B (thus f after g).

The retraction of “f“ would exist if “g“ (or any other morphism B→A) would satisfy the rule: “g“ after “f“ is the identity of A.

The morphism “f“ could be a retraction if “f“ after any morphism ends up being 1B. In fact, as “f“ after “g“ is the identity of B, we can also say that “f“ is a retraction of “g“.

More resources:
As described by Wikipedia (on sections):

A section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : X → Y and g : Y → X are morphisms whose composition f o g : Y → Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g.

Every section is a monomorphism, and every retraction is an epimorphism.

Wikipedia also states that “a monomorphism is an injective homomorphism” and that “a homomorphism is a structure-preserving map between two algebraic structures of the same type”. Also that “Epimorphisms are categorical analogues of surjective functions”. Therefore, sections are injective functions and so they require a left inverse, and retractions are surjective functions and so they require a right inverse.

Now, specifically on retractions, Wikipedia says that: "In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace which preserves the position of all points in that subspace. A deformation retraction is a mapping which captures the idea of continuously shrinking a space into a subspace.".

As a side note, the book of Conceptual Mathematics says that retractions are easy to find once an idempotent map "e" is found to be mapping all of the elements of a larger object into itself, effectively "sorting" it and thus maintaining the order.

On the examples I provided in the original question:
Having sorted out the definitions, it’s clear that the examples in the original question are incomplete, and it’s the result of some confusion.

In the first example,

• All people in A live in the same country B (the codomain B has one element). The verb “live” stands for: f:A→B
• Therefore, A > B.
• The country B has a representative in A, thus: g:B→A
• This one representative makes viable a section from B to A on g, because f:A→B after g:B→A is 1B, the identity on B.
• There is no retraction of “f” because there’s no way to recover the identity of A after we’ve mapped it into B.
• However, there's a retraction between the maps A and B, since every person is related to their country representative by an idempotent map, so the representatives effectively sort the whole population into the same groups of people related to each country.
• This retraction is necessarily f, and it is going to be a retraction of the section g, with a resulting identity on B.
• Given any group of criteria in an object T defining the population A, two relationships of T with A called x1 and x2 don’t need to be the same if we only know that the result of f(x1) = f(x2), because many people from A living in a same country B can be associated with many traits from T.

On the second example:

• A is an object composed of some countries. B is the set of all people, so A ≤ B.
• All people live in countries, so g:B→A.
• There’s a representative per country, so f:A→B.
• The morphism g is a retraction of f as long as g after f is the identity on A, which is true since all people is related to their representatives by country, thus there's an idempotent map in B.
• There is going to be a section of f as long as the amount of people in B has the same size of A (which is valid under A ≤ B), so that f after g is the identity on B. In this case, A and B would be isomorphic.
• The is also a section f of g, because g after f is 1A. The representative lives in the country it represents. Thus f is a section of g as much as g is a retraction of f.
• If there’s a criteria defining A, called T, of which any two maps x1 and x2 are equal after f, so that f(x1) = f(x2), then x1 = x2. Imagine a set of criteria A that define the representatives of every country, criteria mapped to an unknown representative of a known country immediately reveal the representative, for the definition of the section, not because of the criteria themselves.

An example of a similar category of objects without retraction would be one without an idempotent map. For example: if we would compare representatives of a specific political party by state, then even though each representative is related to a state, and every person is related to a state, not every person would be related to the representative of the political party of that state, since not every person necessarily belongs to that political party.